Nature of the matrix elements which describe the

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Unformatted text preview: des of the reaction under study. In almost all cases, we wish to determine the amplitude of the matrix element that allows us to transition from some initial state i (with initial spin Ji ) to some final state f (with final spin Jf ). For a charged current interaction of the type e þ AZ ! eÀ þ AÃZþ1 , the cross N N À1 section can be written in terms of a very general expression X  d Ep X 1 ^ ¼ ee jhfjH W jiij2 ; (40) d cos 2 i ð2Ji þ 1Þ Mi ;Mf where Ee , pe , and cos are the outgoing electron energy, momentum, and scattering angle, respectively, and Ji is the total spin of the target nucleus. The sum is carried over all the accessible spins of the initial and final states. The term in brackets encapsulates the elements due to the hadroniclepton interaction. A Fourier transform of the above expression allows one to express the matrix elements of the Hamiltonian in terms of the four-momenta of the initial and final states of the reaction. The Hamiltonian which governs the strength of the interaction is given by the product of the ~ ~ hadronic current H ðxÞ and the leptonic current J ðxÞ Z GV p ~ CC ~ ~ H CC ¼ F ffiffiffiud ½JCC; ðxÞH ðxÞ þ H:c:Šdx; W 2 GF Z ~ NC ~ ~ H NC ¼ pffiffiffi ½J NC; ðxÞH ðxÞ þ H:c:Šdx; W 2 where CC ~ Æ~ H ðxÞ ¼ V ðxÞ þ AÆ ðxÞ; ~ NC ~ 0~ s H ðxÞ ¼ ð1 À 2sin2 W ÞV ðxÞ þ A0 ðxÞ À 2sin2 W V : ~ We concentrate on the charged current reaction first. In the above expression, the V Æ and AÆ components denote the vector and axial-vector currents, respectively. The Æ and 0 index notation denotes the three components of the isospin raising (lowering) currents for the neutrino (or antineutrino) reaction. The final ingredient V s denotes the isoscalar current. For the case of the impulse approximation, it is possible to write down a general representation of the hadronic weak current in terms of the relevant spin contributions  a F ð q2 Þ a " F1 ðq2 Þ  þ i 2  q hfjV ðq2 Þjii ¼ uðp0 Þ 2mn  2  q þi FS ðq2 Þ uðpÞ; MN  a F " FA ðq2 Þ  5 þ P ðq2 Þq 5 hfjAa ðq2 Þjii ¼ uðp0 Þ  2 MN  F þ T ðq2 Þ q 5 uðpÞ: MN Here a is indexed as a ¼ Æ, 0,  are the spin matrices, " uðp0 Þ and uðpÞ are the Dirac spinors for the target and finalstate nucleon, MN is the (averaged) nucleon mass, and F½S;1;A;2;P;T Š ðq2 Þ correspond to the scalar, Dirac, axial vector, Pauli, pseudoscalar, and tensor weak form factors, respectively. The invariance of the strong interaction under isospin simplifies the picture for the charged current interaction, as both the scalar and tensor components are zero Joseph A. Formaggio and G. P. Zeller: From eV to EeV: Neutrino cross sections . . . FS ðq2 Þ ¼ FT ðq2 Þ  0: (41) In order to proceed further, one needs to make some link between the form factors probed by weak interactions and those from pure electromagnetic interactions. Fortunately, the CVC hypothesis allows us to do ju...
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