Confinement and Chiral Symmetry Breaking

Confinement and Chiral Symmetry Breaking

What happens when Nf N? and we are no longer in the conformal window?

The expectation is that for Nf < N? the coupling is once again strong enough to lead to confinement, in the sense that all finite energy excitations are gauge singlets.

Most of the degrees of freedom will become gapped, with a mass that is set parametrically by ⇤QCD = μe1/2 0 g (μ) . However, there do remain some massless modes.

These occur because of the formation of a vacuum condensate

This spontaneously breaks the global symmetry of the model, known as the chiral symmetry. The result is once again a gapless phase, but now with the massless fields arising as Goldstone bosons.

We will have a lot to say about this phase. We will say it in Section 5.

For pure Yang-Mills, we saw in Section 2.5 that a Wilson line, W [C] = tr P exp i A in the fundamental representation provides an order parameter for the confining phase, with the area law, hW [C]i ⇠ e A , the signature of confinement.

However, in the presence of dynamical, charged fundamental matter – whether fermions or scalars – this criterion is no longer useful.

The problem is that, for a sufficiently long flux tube, it is energetically preferable to break the string by producing a particle-anti-particle pair from the vacuum.

If the flux tube has tension and the particles have mass m, this will occur when the length exceeds L > 2m/ .

For large loops, we therefore expect hW [C]i ⇠ e μL . This is the same behaviour that we previously argued for in the Higgs phase.

To see how they are related, we next turn to theories with scalars.

2.7.3 The Higgs vs Confining Phase

We now consider scalars. These can do something novel: they can condense and spontaneously break the gauge symmetry. This is the Higgs phase.

Consider an SU (Nc ) gauge theory with Ns scalar fields transforming in the fundamental representation. If the scalars are massless, then the gauge coupling runs as 

For Ns < 22Nc , the coupling becomes strong at an infra-red scale, ⇤QCD = ⇤U V e1/2 0 g0 .

It is thought that the theory confines and develops a gap at this scale. We expect no massless excitations to survive.

What now happens if we give a mass m2 to the scalars?

For m2 > 0, we expect these to shift the spectrum of the theory, but not qualitatively change the physics.

For m2 ⇤2QCD , we can essentially ignore the scalars at low-energies and where we revert to pure Yang-Mills.

The real interest comes when we have m2 < 0 so that the scalar condense. What happens then?

Suppose that we take m2 ⌧ ⇤2QCD . This means that the scalars condense at a scale where the theory is still weakly coupled, g 2 (|m|) ⌧ 1, and we can trust our semi-classical analysis.

If we have enough scalars to fully Higgs the gauge symmetry (Ns Nc 1 will do the trick), then all the gauge bosons and scalars again become massive.

It would seem that the Higgs mechanism and confinement are two rather different ways to give a mass to the gauge bosons.

In particular, the Higgs mechanism is something that we can understand in a straightforward way at weak coupling while confinement is shrouded in strongly coupled mystery.

Intuitively, we may feel that the Higgs phase is not the same as the confining phase.

But are they really different?

The sharp way to ask this question is: does the theory undergo a phase transition as we vary m2 from positive to negative? We usually argue for the existence of a phase transition by exhibiting an order parameter which has different behaviour in the two phases.

For pure Yang-Mills, the signature for confinement is the area law for the Wilson loop.

But, as we argued above, in the presence of dynamical fundamental matter the confining string can break, and the area law goes over to a perimeter law.

But this is the expected behaviour in the Higgs phase. In the absence of an order parameter to distinguish between the confining and Higgs phases, it seems plausible that they are actually the same, and one can vary smoothly from one phase to another.

For example: SU (2) with Fundamental Matter

Consider SU (2) gauge theory with a single scalar in the fundamental representation.

For good measure, we’ll also throw in a single fermion , also in the fundamental representation. We take the action to be

Note that it’s not possible to build a gauge invariant Yukawa interaction with the matter content available. We will look at how the spectrum changes as we vary from v 2 from positive to negative.

Higgs Phase, v 2 > 0: When v 2

QCD we can treat the action semi-classically. To read off the spectrum in the Higgs phase, it is simplest to work in unitary gauge in which the vacuum expectation value takes the form h i = (v, 0).

We can further use the gauge symmetry to focus on fluctuations of the form = (v + ̃, 0) with ̃ 2 R. You can think of the other components of as being eaten by the Higgs mechanism to give mass to the gauge bosons.

The upshot is that we have particles of spin 0,1/2 and 1, given by • A single, massive, real scalar ̃. • Two Dirac fermions i = ( 1 , 2 ).

Since the SU (2) gauge symmetry is broken, these no longer should be thought of as living in a doublet. As we vary the mass m 2 R, there is a point at which the fermions become massless.

(Classically, this happen at m = 0 of course.)

• Three massive spin 1 W-bosons Aaμ , with a = 1, 2, 3 labelling the generators of su(2).

Confining Phase, v 2 < 0: When v 2 < 0, the scalar has mass m2 > 0 and does not condense.

We expect to be in the confining phase, in the sense that only gauge singlets have finite energy.

We can list the simplest such states: we will see that they are in one-to-one correspondence with the spectrum in the Higgs phase

• A single, real scalar † . This is expected to be a massive excitation.

If we were to evaluate this in the Higgs phase then, in unitary gauge, we have † = v 2 +v ̃+. . . and so the quadratic operator corresponds to the single particle excitation ̃, plus corrections.

There are further scalar operators that we can construct, including tr Fμ⌫ F μ⌫ and ̄ .

These have the same quantum numbers as † and are expected to mix with it. In the confining phase, the lightest spin 0 excitation is presumably created by some combination of these.

• Two Dirac fermions. The first is 1 = † . The second comes from using the ✏ij invariant tensor of SU (2), which allows us to build 2 = ✏ij i j . If we expand these operators in unitary gauge in the Higgs phase, we have 1 = v 1 + . . . and 2 = v 2 + . . ..

It’s now less obvious that each of these fermions becomes massless for some value of m 2 R, but it remains plausible. Indeed, one can show that this does occur.

(A modern perspective is that the fermionic excitation is in a different topological phase for m 0 and m ⌧ 0, ensuring a gapless mode as we vary the mass between the two.)

• Finally, we come to the spectrum of spin 1 excitations. Since we want these to be associated to gauge fields, we might be tempted to consider gauge invariant operators such as tr F μ⌫ Fμ⌫ , but this corresponds to a scalar glueball.

Instead, we can construct three gauge invariant, spin 1 operators. We have the real operator i † Dμ , and the complex operator ✏ij i (Dμ j ). In unitary gauge, these become v 2 A3μ and v 2 (A1μ + iA2μ ) respectively.

This is a strongly coupled theory, so there may well be a slew of further bound states and these presumably differ between the Higgs and confining phases.

Nonetheless, the matching of the spectrum suggests that we can smoothy continue from one phase to the other without any discontinuity.

We conclude that, for this example, the Higgs and confining phases are actually the same phase.

Another Example: SU (2) with an Adjoint Scalar It’s worth comparing what happened above with a slightly different theory in which we can distinguish between the two phases.

We’ll again take SU (2), but this time with an adjoint scalar field . We’ll also throw in a fermion , but we’ll keep this in the fundamental representation. The action is now

where we’ve now also included a Yukawa coupling between the scalar and fermion.

Once again, we can look at whether there is a phase transition as we vary v 2 .

For v 2 < 0, the scalar field is massive and we expect the theory to be gapped and confine.

Importantly, in this phase the spectrum contains only bosonic excitations.

There are no fermions because it’s not possible to construct a gauge invariant fermionic operator.

In contrast, when v 2 > 0 the scalar field will get an expectation value, breaking the gauge group SU (2) ! U (1), resulting in a gapless photon.

There are also now 2 fermionic excitations which carry charge ± 12 . The spectrum now looks very different from the confining phase.

Clearly in this case the Higgs and confining phases are different.

Yet, because we have fermions in the fundamental representation, we will still have dynamical breaking of the flux tube and so fundamental Wilson loop W [C] does not provide an order parameter for confinement.

Nonetheless, the existence of finite energy states which transform under the Z2 centre of SU (2) – which here coincides with ( 1)F , with F the fermion number – provides a diagnostic for the phase.