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include zero. This is thought to be very closely linked to the property of confinement , which is simply a statement that gluons cannot exist on their own, but only in colourless bound states. Lattice gauge theories have shown to the satisfaction of most that quarkless QCD exhibits such a phenomena, but solving this problem with due mathematical rigour still remains one of the Millennium Prize Problems. Experimentally it is certainly true that there is a mass gap in QCD. 

## 1.2.8 Chiral Symmetry Breaking 

We have already seen in section 1.2.4 that massless QCD anomalously breaks its classical scale invariance. There is a second classical symme- try to the massless QCD Lagrangian: that of chiral symmetry, meaning that the left and right handed quarks transform independently. 

The Lagrangian of massless QCD is given by taking equation (1.1) and simply setting m = 0: 

(1.31) possesses a chiral symmetry which can be seen if we write ψ as    ψ$_{L}$ ψ$_{R}$    and choose the chiral representation of the gamma matrices: 

then (suppressing the gauge field which doesn’t affect the analysis), equation (1.31) becomes 

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