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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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3.7. Nuc1ear forces us<strong>in</strong>g the method of unitary transformation 89<br />

To obta<strong>in</strong> the effective potential via eq. (3.230), we need to know the operators >..a AZ77 that can<br />

be evaluated along the l<strong>in</strong>es described <strong>in</strong> the last section. Let us start with >..1 A077. <strong>The</strong> lead<strong>in</strong>g<br />

contributions to eq (3.206) with i = 1 appear at order r = 0, as follows from eq. (3.216). We can<br />

at lead<strong>in</strong>g order. Concrete, we obta<strong>in</strong> for this equation at r = 0:<br />

(3.240)<br />

now make use of formulae of appendix B to determ<strong>in</strong>e all terms that contribute to the decoupl<strong>in</strong>g<br />

equation (3.206) with i = 1<br />

Bere and <strong>in</strong> what follows, we denote the contribution from the free Bamiltonian Ho by the pert<strong>in</strong>ent<br />

nucleonic and pionic free energies, E and w, respectively. For >..1 A077 we obta<strong>in</strong>:<br />

(3.241)<br />

Proceed<strong>in</strong>g analogously and perform<strong>in</strong>g the recursion <strong>in</strong> the direction <strong>in</strong>dicated <strong>in</strong> fig. 3.5, we f<strong>in</strong>d:<br />

>..4 Ao77<br />

>..1 Al77<br />

>.. 2 Al77<br />

>..4Al77<br />

>..1 A277<br />

o ,<br />

0,<br />

---H277<br />

>..2<br />

-<br />

>..2<br />

w1 + w2 w1 + W2<br />

H1>..1 A077 ,<br />

---- >..4 -- -H277 ,<br />

w1 +W2 +W3 +W4 >..2<br />

---H377 ,<br />

w1 +W2<br />

>..1 >..1 >..1<br />

--H1>.. 2 w<br />

A077 - -H2>..1<br />

w<br />

A077 + -A077H1>..1<br />

w<br />

Ao77<br />

+ -A077H277<br />

>..1 >..1 >..1<br />

- 6' w -A077 w + -Ao77E w ,<br />

>..1 >..1 >..1<br />

--H177 - -H1>.. 2 w w<br />

Al77 - -H3>..1<br />

w<br />

Ao77 .<br />

Bere we have used the fact that the follow<strong>in</strong>g operators vanish:<br />

(3.242)<br />

(3.243)<br />

(3.244)<br />

(3.245)<br />

(3.246)<br />

(3.247)<br />

(3.248)<br />

(3.249)<br />

Note that only those operators >..a AZ77 are shown explicitly <strong>in</strong> eqs. (3.241)-(3.248) that we will<br />

need <strong>in</strong> furt her calculations. In particular, we do not show >.. 3 Ao77 and >..2 A277 that do not vanish.<br />

<strong>The</strong> effective potential def<strong>in</strong>ed <strong>in</strong> eq. (3.230) can be found us<strong>in</strong>g eq. (3.197) and the dimensional<br />

analysis rules (3.223)-(3.229). <strong>The</strong> m<strong>in</strong>imal possible value of 1/ turns out to be 1/ = 6 - 3N.<br />

Correspond<strong>in</strong>gly, the lead<strong>in</strong>g order N-body potential can be written <strong>in</strong> the form:<br />

(6-3N) ( At 1 1A At d A )<br />

Veff = 77 H2 + 0>" H1 + H1>" 0 + 0/\ W 0 77 · (3.250)<br />

Note that <strong>in</strong> the last term only the pionic free energy w contributes at lowest order. No contribution<br />

to the potential appears at order 7 - 3N because of parity <strong>in</strong>variance. This is also clear from the

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