The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory
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derivative. Further helpful equalities are<br />
1);2:1 ,<br />
1);2:-2+p<br />
173<br />
(A.5)<br />
(A.6)<br />
Whereas (A.5) is generally valid s<strong>in</strong>ce no renormalizable <strong>in</strong>teractions are allowed by chiral symmetry,<br />
(A.6) follows from simple <strong>in</strong>spection of eq. (3.211): for n = 0 there are at least two derivatives<br />
and for n 2: 2 it is immediately apparent.<br />
Below we will exam<strong>in</strong>e the chiral power v for all terms enter<strong>in</strong>g the left-hand side of eq. (3.206).<br />
• )..4k +iHK'fl<br />
v = 4 - 3N - 4k - i + I); • (A.7)<br />
This follows from eq. (3.210) if one notes, that now there is one energy denom<strong>in</strong>ator less.<br />
We then obta<strong>in</strong> from eqs. (A.3), (A.4), (A.6):<br />
v 2: 4 - 3N, when k = 0 ,<br />
v 2: 2 - 3N, when k > 0<br />
• )..4k +i HK)..4Q+ j Az'fl<br />
v = 4 - 3N - 4k - i + 2q + j + I); + l .<br />
<strong>The</strong> follow<strong>in</strong>g possibilities are to be considered:<br />
(A.8)<br />
1. q::;k-l<br />
In this case the number of pions at the vertex is restricted by p 2: 4k + i - 4q - j.<br />
<strong>The</strong>refore, the <strong>in</strong>equality (A.6) takes the form<br />
It then follows from eq. (A.8), that<br />
2. q = k<br />
Here we have:<br />
I); 2: -2 + 4k + i - 4q - j<br />
v 2: 4 - 3N - 2k<br />
p 2: Ij - il<br />
(A.9)<br />
(A.I0)<br />
(A.11)<br />
Apply<strong>in</strong>g eq. (A.4) and us<strong>in</strong>g the (obvious) <strong>in</strong>equality j - i 2: -Ij - il one can justify<br />
eq. (A.I0) also for that case.<br />
3. q 2: k + 1<br />
One can easily prove eq. (A.I0) far this case us<strong>in</strong>g eq. (A.5) and not<strong>in</strong>g, that j -i 2: -3.<br />
• E()..4k +i ) .. 4k+i Az'fl<br />
v 2: 4 - 3N - 2k + l 2: 4 - 3N - 2k . (A.12)<br />
This follows immediately from eq. (3.214). Here, the first <strong>in</strong>equality becomes the exact<br />
equality when only the pion k<strong>in</strong>etic energy is taken <strong>in</strong>to account <strong>in</strong> E()..4k +i ).<br />
• )..4k +iAZ'flHK'fl<br />
v = 4 - 3N - 2k + l + I); • (A.13)<br />
<strong>The</strong>re is no non-vanish<strong>in</strong>g operator 'flHK'fl with I); < O. That is why the <strong>in</strong>equality (A.I0) is<br />
aga<strong>in</strong> valid.