The chiral anomaly

0¹. Notice that this symmetry is preserved in the case of degenerate quark masses, so in the case of N_f = 2, given the similar mass of uand d quarks, the symmetry is only mildly broken².

For spontaneously broken continuous symmetries, the Goldstone theorem tells us that the breaking comes accompanied with the appearance of massless Goldstone bosons. For our model of QCD with N_f = 3 massless quark flavours, one expects the presence of 8 Goldstone bosons from the breaking of SU(N_f)_L ×SU(N_f)_R and 1 from the breaking of the U(1)_A symmetry. In real world QCD, with 3 light but not massless quarks, one expects to find light mesons corresponding to the would be Goldstone bosons. One can readily identify some of these particles, the 3 π mesons, the 4K mesons and the η meson. There is still one particle missing in this picture, and as pointed out by Weinberg in Ref. [46], if the non-zero mass of the would be Goldstone boson associated to the breaking of the U(1)_A symmetry had the same origin as the one of the rest of previously mentioned mesons, the mass of the ninth pseudo Goldstone boson should be bounded by the condition m_GB <√

3m_π. However, the lightest pseudoscalar meson compatible with the U(1)_A symmetry is the η⁰ meson, whose mass is about 958 MeV, and is therefore too heavy to be the particle predicted by Weinberg.

3.1.1 The chiral anomaly

The case of the U(1)_A symmetry is special, as it is in fact explicitly broken. The breaking is not observed in the classical Lagrangian, but it is only a consequence of quantum corrections. This explicit breaking is known as the chiral anomaly, and in simple terms, it arises because the fermionic part of the integral measure which enters in the path integral is not invariant under a chiral transformation [47].

The chiral anomaly can be computed perturbatively, as was shown originally in the case of QED by Adler, Bell and Jackiw [48, 49] after considering the diagram coupling the axial current

J_µ5 = ¯Ψγ_µγ₅Ψ, (3.3)

1Note that this is a non-perturbative effect and as such has been an important subject of study of lattice QCD [44, 45]

2This is precisely theSU(2)isospin symmetry observed in the nucleons.

3.1. THE FATE OF THE U(1)_A SYMMETRY 21 to two photons with an intermediate triangle shaped quark loop. For a non-Abelian gauge theory, the anomaly can be computed in a similar way and at one loop the result is

∂_µJ_µ5(x) =− 1

16π² TrF_µν ^∗F_µν = 2q(x), (3.4) where^∗F_µν =_µνρσF_ρσ, andq(x)is the topological charge density. The Adler-Bardeen theorem [50] then guarantees that the anomaly does not get any corrections at higher loop orders. Note that for convenience, the result in Eq. (3.4) has been written in terms of the rescaled fields gA_µ →A_µ and we will keep this convention throughout this chapter unless stated otherwise.

As shown by ’t Hooft [51, 52], the anomaly gets contributions from non equivalent vacuum configurations with different winding number. Moreover, the different con-figurations (topological sectors) are connected by non-perturbative objects known as instantons; which are finite action solutions to the classical equations of motion in Euclidean space time. Their non-perturbative nature is made evident as instantons have a classical action³

S_instanton= 8π²

g² . (3.5)

.

In fact, the integrated topological charge Q Q=

Z

d⁴x q(x) = Z

d⁴x ∂_µK_µ, (3.6)

can be written in terms of the local Chern-Simons current K_µ[53], and using Gauss’

law it can be cast into a surface integral over the boundary Σ of R⁴, which is isomorphic to the three sphere S³, so that

Q= Z

dΣˆn_µK_µ, (3.7)

where nˆ_µ is a normal vector to Σ. This expression shows that Q is a topological quantity that depends on the configuration of the gauge fields in the boundary of the space-time, where the gauge field A_µ(x) approaches a pure gauge field. The expression in Eq. (3.7) is the winding number of the gauge manifold and in the case of SU(N) it is given by the third homotopy group of S³, π₃(S³) = Z. In this sense, the vacuum configurations can be classified according to their integer winding number Q.

3Note that the transition probability between two instanton vacua is give bye^{−Sinstanton} and its smallg expansion is zero at all orders.

22 3.1. THE FATE OF THEU(1)_A SYMMETRY Going back to the computation of the anomaly, a very instructive way to derstand its origin is to look at the transformation properties of the measure un-der a chiral transformation. In the following, we review the discussion presented in Refs. [47, 54]. Let us look at the chiral transformation of the fermion fields

Ψ(x)→e^iα(x)γ5Ψ(x), Ψ(x)¯ →Ψ(x)e¯ ^iα(x)γ5. (3.8) Then, under an infinitesimal transformation, and neglecting the change in the inte-gration measure, the fermionic part of the functional integral changes to

Z →

If the Jacobian of the transformation in Eq. (3.8) was equal to 1, Eq. (3.9) would imply the conservation of the axial current as in the classical theory. Therefore, the anomaly must appear from the change in the integration measure dΨdΨ, which can¯ be evaluated from the Jacobian of the chiral transformation. To do that, one can use a basis{φ_n}of eigenvectors of the hermitian operatorD/ =γ_µD_µ and show that [47]

The sum in Eq. (3.11) has to be regularized as discussed in Ref. [47], and in more detail in Ref. [54]. We do not go into the details and simply quote the final result of the calculation

Finally, combining Eqs. (3.12), (3.10) and (3.9), and taking the variation with respect toα(x) allows us to obtain the chiral anomaly from Eq. (3.4).

A remarkable property connected to the results presented above is that of the Atiyah-Singer index theorem [55]. Naively, using the fact that

γ₅, /D = 0, one can argue that in passing from (3.11) to Eq. (3.12), the only non-zero contribution comes from the zero-modes of D. The famous result then connects the number of/ zero modes with right (n_L) and left (n_R) chirality to the integral of the topological charge density, i.e., the integrated topological charge Q

n_L−n_R =− 1 32π²

Z

d⁴x TrF_µν ^∗F_µν =Q . (3.13)