Introductory Nuclear Physics

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PHY431 Lecture 10 16 4/30/2000 11.5.2 The Real Pion: the Pion Form Factor The real pion is built from quarks, and therefore will not have a simple point-like electromagnetic interaction as represented by the Feynman graph and the transition element (11.57). But although the real pion current j π µ will differ from the point-like “pion” current, it will remain a Lorentz contravariant vector! At the scalar-photon-scalar vertex only three fourvectors are available: p 1 , p 3 , and q = p 1 p 3 , of which only two are independent; choose: q =( p 1 p 3 ) and ( p 1 + p 3 ) . Note that these are perpendicular (in the fourvector sense) because of current conservation, similar as for the point-like pion. Finally, there are only two Lorentz scalars at the vertex: ( p 1 ) 2 = ( p 3 ) 2 = m 2 , and q 2 = ( p 1 p 3 ) = 2 m 2 2 p 1 p 3 of which only one is independent, choose q 2 . The most general current fourvector that can be constructed for the real pion is then: 13 () 22 13 1 3 1 ( ) () ip p x jje F q p pG q q e EE µµ −−  == + +  (11.58) Current conservation further restricts this to a single unknown Form Factor F ( q 2 ): 13 1 3 2 2 2 2 1 3 0 4 0( ) ( ) ( ) ) ( ) ( ) ( ) ( ) ( ) 0 ( ) 0 because (as for a point-like pion): ( ) ( ) ( ) iq x iqx e e jj F q p p G q q e F qpp G q qq e F qqpp G q q G q qp p p p p p p p µπ = =∂ = + + = =+ + + + = = += 2 "real " 1 3 0 1 Thus: ( )( ) iqx mm je F q p p e ππ =−= (11.59) The Form Factor is normalized to 1 at q 2 =0, because the electric charge (or electromagnetic coupling constant) is defined at large distance (zero momentum transfer q ).
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PHY431 Lecture 10 17 4/30/2000 12. The Dirac Equation The Dirac equation caused a revolution in the description of elementary particles. It provides a fully relativistic description of the particles, and incorporates by necessity – and therefore explains – the in- trinsic half-integer spin of fermions. All elementary matter fields are fermionic (leptons and quarks), and the Dirac equation is essential to their description. Historically, Dirac postulated his equation well after the Klein-Gordon equation was invented and abandoned again because of the appearance of negative energy solutions. Searching for a relativistic wave equation with a first order time derivative so as to obtain positive definite probabilities, Dirac postulated: 123 () ii m im tx y z ψ ααα ψβψ α βψ  ∂∂ =− + + + = − ⋅∇+   "! (12.1) with i and β four unknowns to be determined. In order to fix the unknowns, Dirac required that the solutions should also satisfy the Klein-Gordon equation. Thus: 2 22 2 11 33 12 21 1 2 13 31 1 3 23 32 2 3 12 2 2 ( ) (Klein-Gordon equation) ( )( ) (squared Dirac equation) ( ) [ m t m im im ψψ αβαβ αα α β βα −= + =− ⋅∇+ − ⋅∇+ = =− ∇− ∇− ∇ + −+ −+∇ 3 ] im ββ +∇ (12.2) Clearly the unknowns must be matrices, for if the cross-terms are to vanish they must anti-commute, e.g. ( 1 2 + 2 1 ) = { 1 , 2 }= 0. The matrices α , must satisfy: 0 2 1 {, }2 (,)( , ) 0( ) 1 with: { , } 2 {,}0 0 i i ij i j ji i g µ ν µν αβ δ γγ γ αβ βα == =  ≡≡  += =  = =  α (12.3)
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Lecture 10 - PHY431 Lecture 10 16 4/30/2000 11.5.2 The Real...

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