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The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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3.3. <strong>Chiral</strong> perturbation theory with pions<br />

t<br />

u<br />

(Pa - Pe) 2 = (Pd - Pb) 2 ,<br />

(Pa - Pd) 2 = (Pe - Pb) 2 .<br />

71<br />

(3.147)<br />

For simplicity, we will consider the chiral limit with Mn = O. <strong>The</strong> Mandelstarn variables then<br />

satisfy 82 + t2 + u2 = O. <strong>The</strong> S-matrix element can be expressed <strong>in</strong> the form<br />

<strong>The</strong> transition amplitude M for the pion-pion scatter<strong>in</strong>g can be parametrized as<br />

(3.148)<br />

(3.149)<br />

<strong>in</strong> terms of a s<strong>in</strong>gle function A. To lead<strong>in</strong>g order (v = 2) this function is completely determ<strong>in</strong>ed<br />

by the <strong>in</strong>ter action term<br />

_ 1 _(1r . 8 1r)(1r . 8J11r)<br />

21; J1 , (3.150)<br />

correspond<strong>in</strong>g to the lead<strong>in</strong>g order Lagrangian (3.136), see eq. (3.106). Note that the chiral<br />

symmetry fixes the coefficient 1/(2{;) for this term. For the function A one obta<strong>in</strong>s<br />

A (2) (8,t,U) _- � J; . (3.151)<br />

As shown <strong>in</strong> fig. 3.2, at next-to-Iead<strong>in</strong>g order one has to evaluate the one-Ioop diagram with both<br />

vertices (3.150) as weH as the tree graph with the <strong>in</strong>teractions<br />

(3.152)<br />

Here we use the notation of reference [133] . Clearly, the coupl<strong>in</strong>gs q, c� are some l<strong>in</strong>ear comb<strong>in</strong>ations<br />

of the LECs def<strong>in</strong>ed <strong>in</strong> eq. (3.141). For the function A one f<strong>in</strong>ds at next-to-Iead<strong>in</strong>g<br />

order<br />

where A is the ultraviolet cut-off. Introduc<strong>in</strong>g the renormalized coupl<strong>in</strong>gs<br />

allows to express the amplitude <strong>in</strong> the form<br />

1 ( 1 2 ( 8 ) 1 2 2 2 ( t )<br />

-- - - 8 In - - -- (U - 8 + 3t ) In -<br />

(21 n)4 27r2 /12 127r2 /12<br />

R R' 1 2 2 2 ( U ) C4 C4 2 2 2 )<br />

- -- (t - 8 + 3u ) In - - -8 - - (t + U ) .<br />

127r2 /12 2 4<br />

(3.153)<br />

(3.154)<br />

(3.155)

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