Introduction to the Standard Model of Particle Physics

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Part II Kinematics 39 10/23/2000 Part II Kinematics 1 Special Relativity Before going any further, we present a quick overview of notations for the kinematics of spe- cial relativity. 1.1 Four Vectors and Lorentz Transformations Four-vectors are denoted by x µ , with the greek index =0,1,2,3; a single time-like component 0, followed by the three space-like components 1,2 and 3: x = ( x 0 , x 1 , x 2 , x 3 ). The space-like indices 1,2, and 3 are habitually represented by lower case roman characters: e.g. x i = x = ( x 1 , x 2 , x 3 ). In (special) relativity a space-time distance (four-length) t 2 x 2 y 2 z 2 is constant and invariant under Lorentz trans- formations: i.e. t 2 x 2 y 2 z 2 = t 2 x 2 y 2 z 2 , a direct consequence of the constancy of the speed of light for all observers. The space-time fourvector coordinate is denoted by x = ( x 0 , x 1 , x 2 , x 3 ) = ( x 0 , x i ) = ( t , x ) = ( t , x , y , z ). The space-time components mix between themselves by Lorentz transformations from one inertial sys- tem to another. A Lorentz transformation from an (unprimed) system to another (primed) system mov- ing at relative velocity β// x with respect to the first, can be expressed in terms of speed β and relativis- tic factor γ (1 2 ) ½ as: 22 2 00 01 , with det( ) 1 10 µµ νν γβ βγ   Λ= Λ = − =  (II.1) Note the lower and upper indices: this facilitates the definition of a summation convention of same in- dices: 3 01 2 3 0123 0 1 2 3 0 ' xx x x x x x t x y z ν µ ν µµµ = Λ (II.2) The invariance of the length of x is indeed satisfied: t 2 x 2 y 2 z 2 = ( t z ) 2 x 2 y 2 – ( t + z ) 2 = ( 2 2 2 ) t 2 x 2 y 2 – ( 2 2 2 ) z 2 = t 2 x 2 y 2 z 2 . This leads to a definition of an invariant four-dimensional dot-product: x x = ( x 0 ) 2 – ( x 1 ) 2 – ( x 2 ) 2 ( x 3 ) 2 = g µν x x , where the metric tensor g in special relativity is defined as: 1 000 00 10 00 0 1 gg == (II.3) Other commonly used fourvectors, contravariant (= index high) or covariant (= index low), are: ( , ) (, ) , ( ,) , ( , ( , ) , ( , ) pE i ii j A tt t ρφ ∂∂ = = −∇=∂ ∂= ∇ ∂= + = = p j A ## # (II.4)
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Part II Kinematics 40 10/23/2000 1.2 Lorentz Scalars The length of a Lorentz fourvector is invariant under Lorentz transformations, as a special case of the invariance of the Lorentz fourvector dot product. The four-momentum transfer (= difference) q is defined as q p 3 p 1 , where the quantities q and p are implied fourvectors. The labels 1 and 3 are particle labels, with the labeling convention shown in the accompanying Figure. Note that the four-momentum transfer q is defined using external lines only; i.e. it is independent of the details of the interaction taking place in the volume. The Lorentz scalar q 2 is related to the properties of particles 1 and 3 only: 22 2 31 3 1 3 1 3 1 3 1 3 1 3 1 3 1 1 3 () ( ) 2 22c o s qq q p ppp pp p p mm E E E E µµ µ θ ≡= = + =+− + + (II.5) which is correct in any reference system. e.g. in the center of mass system (CMS), defined by p 1 + p 2 =0, and in the special case that m 1 = m 3 , equation (II.5) simplifies to:
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This note was uploaded on 02/05/2008 for the course PHY 557 taught by Professor Rijssenbeek during the Fall '00 term at SUNY Stony Brook.

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Lecture 07 - Part II Kinematics 39 10/23/2000 Part II...

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