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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3.1. Chira,l symmetry<br />
, ,<br />
:: - =:\<br />
I \<br />
... ... ... �--- ---"', ...<br />
/.. ,;<br />
.. ,<br />
,<br />
Figure 3.1: Spontaneous break<strong>in</strong>g of the symmetry.<br />
Here, we do not sum over the values of a. Perform<strong>in</strong>g a space-time translation and us<strong>in</strong>g the fact<br />
that the vacuum state is translational <strong>in</strong>variant we can rewrite eq. (3.43) as follows:<br />
This matrix element has to be different from zero and hence is <strong>in</strong>f<strong>in</strong>ite.<br />
53<br />
(3.44)<br />
Spontaneous breakdown of a cont<strong>in</strong>uous symmetry implies, accord<strong>in</strong>g to the Goldstone theorem<br />
[61], [62], the existence of massless sp<strong>in</strong>less particles called Goldstone bosons. <strong>The</strong> number of<br />
Goldstone bosons equals the number of the broken generators. <strong>The</strong> (abstract) generator ra of<br />
a symmetry group is broken if the correspond<strong>in</strong>g symmetry charge Qa does not annihilate the<br />
vacuum as shown <strong>in</strong> eq. (3.42). We will now sketch the proof of the Goldstone theorem given by<br />
Guralnik et al. [155] and show the presence of the massless particles <strong>in</strong> the physical spectrum. A<br />
very important po<strong>in</strong>t is that there exists some field operator 'l/Ji such that its vacuum expectation<br />
value does not vanish:<br />
(3.45)<br />
We can illustrate the existence of such an operator with a simple classical example shown <strong>in</strong><br />
fig. 3.1. <strong>The</strong>re, the potential V is given as a function offields () and 7r. <strong>The</strong> potential is rotationally<br />
<strong>in</strong>variant. <strong>The</strong> po<strong>in</strong>t (() = 0, 7r = 0) does not correspond to a m<strong>in</strong>imum of the potential. In fact,<br />
there is an <strong>in</strong>f<strong>in</strong>ite number of po<strong>in</strong>ts ( (), 7r) for which the potential takes the m<strong>in</strong>imal value. All<br />
those po<strong>in</strong>ts underlie the requirement<br />
(3.46)<br />
where A i- 0 is some constant. <strong>The</strong> physical vacuum <strong>in</strong> that classical picture corresponds to one