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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54 3. <strong>The</strong> derivation of nuclear forces from chiral Lagrangians<br />
of those po<strong>in</strong>ts, say, to (0" = A, 7r = 0). 5 Thus, the vacuum is characterized by some non-zero<br />
value of the field 0".<br />
Let us now co me back to the Goldstone theorem and consider the vacuum expectation value of the<br />
operator 'l/Ji (x) . U s<strong>in</strong>g the translational properties of field operators and assum<strong>in</strong>g translational<br />
<strong>in</strong>variance of the vacuum, we obta<strong>in</strong><br />
(3.47)<br />
where TJi is some (non-zero) constant. Assum<strong>in</strong>g furt her that -tfjTJj = K,f i- 0 if some of the 7]'S<br />
do not vanish, we obta<strong>in</strong> us<strong>in</strong>g eq. (3.31)<br />
K,f (OI[Qa, 'l/Ji(O)]IO) (3.48)<br />
I d3 x (OI[Jö(x,<br />
t), 'l/Ji(O)]IO)<br />
I: I d3 x {(OIJö(O)e-iP,xln)(nl'l/Ji(O)IO) - (OI'l/Ji(O)ln)(nleiP.X Jö(O)IO)}<br />
n<br />
I:(27r)38(Pn) {(OIJö(O)ln)(nl'l/Ji(O)IO)e-iEnt - (OI'l/Ji(O)ln)(nIJö(O)IO)eiEnt}<br />
n<br />
i- 0,<br />
where we have <strong>in</strong>serted a complete set of <strong>in</strong>termediate energy eigenstates Ln In)(nl and used the<br />
translational <strong>in</strong>variance of the vacuum. Here Pn and En denote the momentum and energy of the<br />
state In). Note that Qa should correspond to a broken generator because otherwise the right-hand<br />
side of eq. (3.48) vanishes. S<strong>in</strong>ce K,f does not depend on time, differentiat<strong>in</strong>g the equation (3.48)<br />
with respect to time leads to<br />
I:(27r)38(Pn)En {(OIJö(O)ln)(nl'l/Ji(O)IO)e-iEnt + (0 I 'l/Ji (0) In) (nIJö(O)IO)eiEnt} = 0 . (3.49)<br />
n<br />
<strong>The</strong> positive and negative frequency parts can not mutually cancel. <strong>The</strong>refore each of both terms<br />
<strong>in</strong> the square bracket <strong>in</strong> eq. (3.49) must vanish for all states In) except those where En = 0<br />
for Pn ---t O. Furthermore, the latter ones must exist s<strong>in</strong>ce otherwise the equation (3.48) will<br />
not be satisfied. Thus, the physical spectrum should conta<strong>in</strong> massless states In) of sp<strong>in</strong> zero<br />
and the same parity and <strong>in</strong>ternal quantum numbers as Jö, whieh are called Goldstone bosons.<br />
For a more rigorous and mathematically correct proof of the Goldstone theorem the reader is<br />
referred to the orig<strong>in</strong>al publications [61], [62] as well as to the references [155], [156] and [157] .<br />
<strong>The</strong> plausible physical <strong>in</strong>terpretation of the Goldstone theorem can be obta<strong>in</strong>ed from the simple<br />
classieal example considered above and illustrated <strong>in</strong> fig. 3.1. <strong>The</strong>re, the vacuum state corresponds<br />
to the po<strong>in</strong>t (0- = A, 7r = 0), whieh is clearly not <strong>in</strong>variant under rotations <strong>in</strong> the 0"-7r surface. <strong>The</strong><br />
potential V (0", 7r) takes its m<strong>in</strong>imum value for the po<strong>in</strong>ts ly<strong>in</strong>g on the r<strong>in</strong>g (j2 + 7r2 = A 2• Thus, no<br />
additional energy is needed to move from one such po<strong>in</strong>t to another. <strong>The</strong> modes correspond<strong>in</strong>g<br />
to such transformations can be identified with the Goldstone bosons.6<br />
Let us now co me back to the QCD Lagrangian. To decide whether the physieal vacuum is chiral<br />
symmetrie or not we first note that the charges of SU(Nf)A carry negative parity. This is because<br />
the ')'5 matrix enters the expression (3.21) for the axial-vector current Jc;t. Thus, if the chiral<br />
symmetry would be realized <strong>in</strong> the Wigner mode, i. e. if the vacuum would be symmetrie, one<br />
5In pr<strong>in</strong>ciple, the vacuum could also be given by a l<strong>in</strong>ear comb<strong>in</strong>ation of those po<strong>in</strong>ts. This possibility can,<br />
however, be excluded for quantum fields <strong>in</strong> a space of <strong>in</strong>f<strong>in</strong>ite volume [133].<br />
GIn this particular example such a mode can be def<strong>in</strong>ed us<strong>in</strong>g the polar angle <strong>in</strong> the surface CJ-7r.