Nuclear and Particle Physics An Introduction

Spring  Semester  2012  Farid  Ould-­‐Saada  

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B.1 Lorentz transformations and 4-Vectors B.2 Frames of references B.3 Invariants

  Particle  of  rest  mass  m,  velocity  u  in  coordinates  (t,x,y,z)  in  frame  S  

  In  S’  moving  with  speed  v=βc  in  z-­‐direction  coordinates  in  S’:  (t’,x’,y’,z’),  u’    

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x '= xy '= y

z'= γ z − vcct

⎛

⎝ ⎜

⎞

⎠ ⎟

ct'= γ ct − vcz

⎛

⎝ ⎜

⎞

⎠ ⎟

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

u'= u − v

1− uvc 2

⎧

⎨ ⎪

⎩ ⎪

β ≡vc

γ =11− β2

: Lorentz factor

γ(u') =1

1− u'c⎛

⎝ ⎜

⎞

⎠ ⎟ 2

= γ(u)γ(v) 1− uvc 2

⎡

⎣ ⎢ ⎤

⎦ ⎥

v

y’

x’

z’

x

y

z S

S’

  Time  dilatation    Distance  contraction  

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 Time  and  space  coordinates  make  up    a  4-­‐vector  

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Lorentz transformations

The space-time and the energy-momentum 4-vectors result in

The Lorentz-transformation of both space-time and momentum-energy four-vectors can be expressed in matrix form:

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  Laboratory  system  (LS)  and  Centre  of  mass  system  (CMS)      In  LS    moving  projectile  a  in  a  beam  strikes  a  target  particle  b  at  rest  

  In  CMS  

  4-­‐vectors  in  both  systems  (L=laboratory,  T=target,  B=beam)  

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Comparison  of  fixed  target  and  colliding  beam  accelerators  

  Unless  u~c  and  cosθC~-­‐1,  final  state  particles  emitted  in  narrow  cone  about  beam  direction  in  LS.  Similarly  with  decays.  

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€

B(EL , p L )+ T(mT2, 0 ) →P(E, q ) + ...

ScatteringangleθL in LS and θC inCMS p L = (0,0, pL ) ; q = (0,qsinθL ,qcosθL )In CMS : p B

' + p T

' = 0

tanθL =1γ(v)

q' sinθC

q'cosθC +vE'c 2

E '= mPc 2γ(u) q'= mP uγ(u) u : velocity of P in CMS

v = pLc 2 EL + mTc 2( )−1 HE:EL ≈ pL c >>mB c 2,mT c 2

⎯ → ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ ⎯ v ≈ c(1−mTc/pL) ≈ c

γ(v) ≈ pL

2mTc⇒ tanθL ≈

2mTcpL

⋅usinθC

ucosθC + c

  Laboratory  system  (LS)  and  Centre  of  mass  system  (CMS)    

  More  efficient  to  work  with  quantities  that  are  invariant!  

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InLS : pL =c2mT

s − (mT +mB )2[ ] s − (mT −mB )

2[ ]

InCMS :p =c2 s

s − (mT +mB )2[ ] s − (mT −mB )

2[ ]

 Invariant  under  all  permutations  of  its  arguments  

Minimum Laboratory energy to produce particle M

  Mass  of  decaying  particle  =  invariant  mass  of  decay  products  

  Dalitz  plots  

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- Crystal barrel at LEAR (Low Energy Antiproton Ring, CERN Meson spectroscopy - Plot with high degree of symmetry: 3 identical particles - Clear enhancements due to resonances

  Mandelstam  variables  

  Rapidity  and  pseudo-­‐rapidity  

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A + B→C +D

s = pA + pB( )2/c 2 t = pA − pC( )2

/c 2 u = pA − pD( )2/c 2

• t + t + u = m j2

j=A ,B ,C ,D∑

• elasticscattering, p,θ in CMSrelative toparticle A,⇒ t = −2p2 1− cosθ( ) /c 2

€

Rapidity :y =12ln E + pL

E − pL

⎛

⎝ ⎜

⎞

⎠ ⎟

Pseudo - rapidity :η = −ln tan θ2⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

http://en.wikipedia.org/wiki/Pseudorapidity

http://en.wikipedia.org/wiki/Mandelstam_variables

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  Pages  358-­‐359  (see  next  2  pages)    B1-­‐B10  

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