From the previous discussion in this work it should become clear that chiral sym-metry plays an crucial role in QCD. In lattice QCD calculations chiral symsym-metry is even more important but much more difficult to preserve. Chiral symmetry in QCD is only exact in the limit of zero quark masses, however the masses of up-and down- quarks are far smaller than the QCD scale. Therefore it should be a good low energy approximation to assume zero quark masses. In this limit the fermionic part of the QCD Lagrangian simplifies to
LF =
nf
X
f
¯
qfiγµDµqf. (2.91)
As next we decompose the quark fieldsqinto left- and right-handed components qL=γLq, qR=γRq, q =qL+qR
¯
qL=¯qγR, q¯R= ¯qγL, q¯= ¯qL+ ¯qR
(2.92)
with
γR= 1+γ5
2 , γL= 1−γ5
2 . (2.93)
After inserting the decomposed spinors into the Lagrangian we obtain L=
nf
X
f
q¯LfiγµDµqLf + ¯qfRiγµDµqfR
, (2.94)
i.e., the Lagrangian decouples into two parts for left- and right-handed quarks resulting inU(nf)L⊗U(nf)Rchiral symmetry.
Due to the quantum anomaly in the axialU(1)A symmetry the symmetry of the quantum field theory is then reduced toSU(nf)L⊗SU(nf)R⊗U(1)Bwhere theU(1)B =U(1)L=Rsymmetry represents baryon number conservation.
2.6.1 The Axial Anomaly and the Atiyah-Singer Index Theo-rem
As a consequence of theU(1)Aquantum anomaly the flavour-singlet axial current jµ5(x) = ¯qγµγ5q, (2.95) which is classically conserved has a non-zero divergence in the quantum field theory
∂µjµ5 =− nf
32π2µνωρtr [Fµν(x)Fωρ(x)] (2.96) due to topological effects in the theory. In particular the axial charge Q5(t) = R d3xj05(~x, t)is related to the topological charge from eq. (2.17) by
Q5(t=∞)−Q5(t=−∞) =nfQ. (2.97) Furthermore the axial anomaly is also deeply connected with the Atiyah-Singer index theorem, which relates the zero-modes of the massless Dirac operator to the topological charge. Since the eigenvalues of the massless Dirac operator γµDµ
are purely imaginary and come in complex conjugate pairs the zero eigenvalues are the only ones which are not paired. The eigenvectors of the zero modes have a definite handiness because the massless Dirac operator anti-commutes withγ5. The Atiyah-Singer theorem states that
index (γµDµ) = nL−nR=nfQ (2.98) i.e., the difference between the number of the left- and right-handed zero-modes is proportional to the topological charge.
2.6.2 Spontaneous Chiral Symmetry Breaking
Although chiral symmetry is only approximate in QCD, we can still expect al-most degenerate states in the spectrum of strongly interacting particles. Since the masses ofupanddownquarks are far below the QCD scale, such an approximate symmetry is observed in nature and hadrons can be classified as isospin multiplets.
Adding the strange quark, the symmetry becomes more approximate but is still visible in the spectrum. Furthermore one observes very light pseudo-scalar par-ticles, the three pions π±, π0, and somewhat heavier pseudo-scalar particles, the four kaonsK±, K0,K¯0 and theη-meson. Thus, it leads us to the indication that chiral symmetry must be spontaneously broken and the observed pseudo-scalar particles must be the corresponding Goldstone bosons.
According to the Goldstone theorem the number of the massless particles is given by the number of the generators of the full symmetry group minus the num-ber in the unbroken subgroup. Here the full chiral symmetry group is
Gχ=SU(nf)L⊗SU(nf)R⊗U(1)B, (2.99) which is broken down to
Hχ =SU(nf)L=R⊗U(1)B. (2.100) Hence we expectn2f−1massless Goldstone bosons. Assuming only approximate symmetry forupanddownquarks we have then three massless Goldstone bosons which can be identified with the three light pions. Fornf = 3we have then addi-tionally the kaons and theη-meson, alltogether eight massless Goldstone bosons.
Of course the pions, kaons and η-meson are not really massless since the chiral symmetry is explicitly broken by the quark masses. But the breaking is relatively small so that we can still identify the Goldston bosons.
In contrast to the electro-weak spontaneous symmetry breaking due to the Higgs field the chiral symmetry breaking has not been derived analytically yet from QCD Lagrangian and its origin remains mysterious. However, in lattice QCD, which is a nonperturbative formulation of QCD, it was shown4 that the chiral symmetry is indeed spontaneously broken [80, 81] in the strong coupling limit. The numerical simulations in lattice QCD confirm this result even in the weakly coupled regime by observing the chiral condensate
χ=h¯qqi, (2.101)
which is an order parameter of the chiral symmetry breaking. If the chiral symme-try is restored the chiral condensate is invariant under chiral rotations and would vanish, otherwise in the chiral broken sector the chiral condensate has a non-zero expectation value.
4In the proof the authors used staggered fermions for the fermionic part of the action, which is controversial due to not yet completely understood properties.
2.6.3 Low-Energy Effective Theory
Since the pions are the lightest particles in QCD they dominate the dynamics of the strong interaction at low energy. A low-energy effective description of QCD is provided by the Chiral Perturbation Theory (χPT). Therein the pion dynamics is predicted by a systematic expansion in powers of external momenta and quark masses. Since pions are Goldstone bosons they are described by fields in the coset spaceU(x)∈Gχ/Hχ=SU(nf).
The extension of the theory to non-zero baryon numbers is a non-trivial task.
An overview on this topic can be found in [82, 83], while here we give only a sketch how this can be done. Extending the chiral perturbation theory to sectors with non-zero baryon number one includes the baryons in the form of a Dirac spinor fieldN(x)andN¯(x)that transforms as anSU(2)Iisospin doublet. Global chiral rotationsL⊗R ∈ SU(2)L⊗SU(2)R can be realised then nonlinearly on this field. In order to realise a chirally invariant action one introduces an SU(2) flavour “gauge” field
vµ(x) = 1