The Nucleon-Nucleon Interaction in a Chiral Effective Field Theory

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