A special reference frame is the center of mass or zero momentum system frame. It is very useful when discussing high energy particle reactions.

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1 High nergy Partile Physis A seial referene frame is the enter of mass or zero momentum system frame. It is very useful when disussing high energy artile reations. We onsider a ollision between two artiles with rest masses m and m. We assume that artile is moving with veloity u r in the laboratory system and that artile is at rest in that system. We have the energy-momentum 4-vetors and the total energy-momentum s s,,, and,,, s s s P + +,,, In a new frame moving along the x-axis with seed V we have V + P' Γ, P', P' 3 / V where Γ. In the enter of mass system, V V CM + V CM and P' r. This says that The energy available for hysial roesses suh as the rodution of new artiles or inelasti events is the total energy in the enter of mass system, '. In the enter of mass system the total energymomentum 4-vetor is ',,, We an find ' by using the fat that the norm of the energy-momentum 4-vetor is invariant ' + or m 4 We have Therefore ' 4 m + + u γm and m, γ ( γ ) m + m ( ) / total energy in laboratory system

2 and / ' ( m + m + γmm ) The fration of energy available for hysial roesses is ' ( ) m + m + γmm γm + m / For the seial ase m m m we have ' + γ At low veloity or low energy of the inident artile (the one that is moving), we have ' γ all energy available In this ase, most of the energy is rest energy and kineti energy is unimortant. In the high seed or high energy limit we have ' m + m / Thus, the useful fration of energy dereases as. For examle, in a 3 GeV aelerator ( GeV ev 6x. J. 6x J) an aelerated roton ( m GeV ) olliding with a hydrogen (rotons) has ' 3. 8 or only 5 GeV is available for reations!!! We will show how to fix this u shortly. Let us look at rodution reations in another way. Suose that we have two artiles that interat with eah other(one is at rest -- the ) and rodue final artiles. The high energy available from the inident artile is onverted into mass of newly reated artiles. We ask the question: What is the minimum energy needed by the inident artile in order to rodue the final state of artiles? In the initial state we have in in, in,, m,,, m,,, + ( ) m in in in In the final state we have

3 i i, i r i where i i + mi 4, i,,3,4,..., ow, the norm of the energy-momentum 4-vetor is invariant in time and aross different frames. Therefore norm in laboratory before norm in enter of mass after This gives icm in i + m icm, r -, i r By definition, however, icm,. After some algebra we have i in icm min m m, i ( ) ( ) This is a minimum when icm, i is a minimum or when icm, i i or all the artiles are at rest in the enter of mass system after the ollision (what are they doing in the laboratory system). Therefore the minimum energy needed by the inident artile (this is alled the threshold energy) is in, threshold m ( ) i in ( ) i i m m m m For examle, onsider the reation π + π + π where a roton is inident on another roton roduing two rotons and three i mesons. The threshold energy is, threshold ( ) m + 3mπ m m m + 6mπ + Clearly, this is a very non-intuitive answer!!! ow let us onsider the differene between a artile aelerator where one artile is aelerated and ollides with a seond artile at rest (as abovelaboratory system) and two artile aelerators 9 m m π

4 where eah artile is aelerated in the same way (olliding beamsenter of mass system). We have Single Aelerator Colliding Beams total lab r m, r total lab,, m total m r, total m,, energy of eah artile In the first ase the aelerator must rodue energy and in the seond ase eah aelerator must rodue energy. The two aelerators are equivalent (same energy available for hysial roesses) if + m Algebra gives the result r,, 4 4 m + m + m If we onsider the ase of very high energy aelerators where >> m we have i m Suose we want to build a single TeV aelerator (TeV 3 GeV ) so 4 that GeV. This is very diffiult to design and requires the develoment of signifiant new equiment ($$$$$$$). If instead we build two smaller aelerators and use them in the olliding beams onfiguration, then we get the same available energy with 5 7 GeV whih we already know how to build. In fat, if we use an old single aelerator of this size that already exists, we then only have to build one small new aelerator ($$). High nergy Collisions arlier we disussed low energy ollisions between artiles using onservation of energy and momentum. Let us look at the same roesses at high energy. We onsider a ollision in whih the inident artile has zero rest mass (hoton) and the artile is at rest. If the artile is an eletron, then this if the so-alled Comton ffet. The roess looks like

5 y o m o x θ φ The hoton momentum is. After the ollision the hoton is sattered through an angle θ with energy and the eletron reoils at an angle φ with veloity r u. The final eletron energy is m e u m γ ( ) u Conservation of energy gives + m + e. Conservation of momentum gives ( x and y diretions) u where osθ + osφ sinθ sinφ r r γm u or + m e 4 We want to eliminate referene to the eletron and find the new hoton energy(that is what is deteted in the exeriment). osθ + osφ osφ osθ os φ osθ sinθ sinφ sinφ sinθ sin φ sin θ Adding these equations we get 4 m osθ + e Using the energy onservation equation we have (after algebra) + ( os θ) m The first thing to note is that >. This means that a free eletron annot absorb a hoton omletely; there will always be a sattered hoton of some energy. If we onvert to wavelengths using

6 we get hν h λ h λ λ θ m ( os ) The shift in wavelength at a given angle is indeendent of the inident hoton energy. You will do this exeriment in Physis 4.