Symmetry, Confinement, and the Higgs Phase

Now let $F(U)=0$ be any physical gauge condition imposed on spacelike links, separately on each timeslice. We will call this an $F$-gauge. Examples include Coulomb gauge, the Laplacian version Vink and Wiese (1992) of Coulomb gauge, and axial gauge. We can prove two theorems relating $\langle\phi\rangle_{F}$ in physical $F$-gauges to our spin wave order parameter $\langle\Phi\rangle$.

$\langle\Phi\rangle$ can be evaluated numerically by lattice Monte Carlo, and for practical purposes one can dispense with the complicated term in (8) proportional to $h$. In that case (8) simplifies to

Once we put the expressions in this form, we can forget about the spin glass analogy and simply regard the calculation as evaluating the vacuum expectation of a gauge invariant operator $\Phi(U)$, which depends only on spacelike links on the $t=0$ timeslice, and which happens to have an involved definition which itself involves a path integral. Then the order parameter can be computed with no difficulty at any $\beta,\gamma$, via a “Monte Carlo within a Monte Carlo” procedure. Figure 2(a) shows how this is done. The procedure is to update both the link and scalar field variables as usual, for e.g. 100 sweeps, generating configurations drawn from the usual probability distribution $e^{-S}/Z$. But computation of the order parameter at each data taking “sweep” (better to call it an “event”) is also done by Monte Carlo simulation. Each data taking event involves $n_{sym}$ sweeps over the lattice, where the spacelike link variables on one time slice, say $t=0$, are held fixed, while all other field variables are updated. During these $n_{sym}$ sweeps we evaluate the average value $\overline{\phi}(\mathbf{x};U)$ at each site on the timeslice, and from there we compute $\Phi[U]$ as in defined in (22). Averaging over many of these data taking events, with different $U_{i}(\mathbf{x},0)$ held fixed, gives us an estimate for $\langle\Phi\rangle$ using $n_{sym}$ sweeps in each procedure. On general statistical grounds

Having computed $\langle\Phi\rangle_{n_{sym}}$ at a variety of $n_{sym}$ values, the last step is to extrapolate the data to $n_{sym}=\infty$, as shown in Figure 2(a). At couplings inside the unbroken phase, the data extrapolates to zero. In the broken phase $\langle\Phi\rangle_{n_{sym}}$ extrapolates to a non-zero value.¹¹1Of course, on a finite lattice, the order parameter always vanishes for $h=0$, and what that means numerically is that without the explicit term breaking term proportional to $h$, for $n_{sym}$ large enough, eventually the data points would depart from the extrapolated line derived from the lower $n_{sym}$ values, and fall to zero. But this departure towards zero, in the broken phase, would occur at ever larger values of $n_{sym}$, as the lattice volume is increased. One can estimate the point at which the transition from a zero to non-zero value occurs, and this is the transition point. The figure shows some sample extrapolations in SU(2) gauge Higgs theory at several $\gamma$ values, on a $16^{4}$ lattice, with $\beta$ is held fixed at $\beta=1.2$. By this procedure, one arrives at a transition line (green data points) shown in Figure 2(b). The upper line is the transition line for spontaneous breaking of the “remnant” symmetry which is left unfixed in Coulomb gauge. This line lies above the breaking of custodial symmetry determined by the order parameter $\Phi$, in conformity with Theorem 1, i.e. custodial symmetry may be broken where remnant symmetry in an F-gauge (such as Coulomb) is unbroken, but not the other way around.²²2One should keep in mind that $\langle\phi\rangle_{F}$ is really the expectation value of a highly non-local quantity, namely $\langle g_{F}(\mathbf{x};U)\phi(x)\rangle$, where $g_{F}$ is the gauge transformation to the F-gauge.

The gauge invariant order parameter $\Phi$ gives us an unambiguous criterion for the breaking of custodial symmetry, which includes at a minimum the global center symmetry of the gauge group. We have suggested that the phase of broken custodial symmetry is the Higgs phase, but that only makes sense if there is a genuine physical distinction which corresponds to this criterion. One might object on a priori grounds that the fact that the confinement and Higgs phases are not entirely isolated from one another by a thermodynamic transition Osterwalder and Seiler (1978); Fradkin and Shenker (1979); Banks and Rabinovici (1979) already rules out any such essential distinction, but one should be a little wary of this argument. We already know of examples where there exist distinct phases of many-body systems which are not separated by a thermodynamic transition. One example is the roughening transition in Yang-Mills theory. In the rough phase, the width of flux tubes grows logarithmically with quark-antiquark separation, and the static quark potential contains a $1/r$ term of stringy origin. Neither of these features hold outside the rough phase. Another example is the Kertesz line in the Ising model in an external field Kertesz (1989), which has to do with a percolation transition in the random cluster formulation of the model. The point here is that analyticity of a local observable like the free energy does not rule out non-analytic behavior in non-local observables, and such observables can be physically important. So our next task is to explain what, precisely, is the physical distinction between the broken and unbroken phases of custodial symmetry, and why these deserve to be called the Higgs and confinement phases respectively.

On the lattice, $E_{V}(R)$ is determined from the lattice logarithmic time derivative

where the timelike link variables arise after integrating out the static quark fields. In the Higgs phase, according to the previous theorems, there is always an $F$-gauge in which $\langle\phi\rangle_{F}\neq 0$. This also implies $\langle U_{0}\rangle\neq 0$. Let $g_{F}(\mathbf{x};U)$ be the gauge transformation to that gauge, and choose

Then evaluating $E_{V}(R)$ in that $F$-gauge we have

which demonstrates the absence of S${}_{\text{c}}$ confinement in the spin glass phase.

Now let us define

and consider the $\mathbf{y}\rightarrow\infty$ limit. This leaves an isolated charged fermion at $\mathbf{x}$ in the charged state, and a neutral hadron at $\mathbf{x}$ in the neutral state. Let $V=V_{F}$ and evaluate the overlap in the Higgs phase

This means there is no essential distinction between the states we have labeled “charged” and “neutral.” That is a consequence of broken global $Z_{N}$ gauge symmetry, with the corollary that the vacuum is not an eigenstate of zero $Z_{N}$ charge.

An isolated color charged particle is the source of a long-range color electric field, and this is ruled out if the theory is massive. In the absence of a massless sector, both the Higgs and confinement phases are C confining, and we have just established that the Higgs phase is not S${}_{\text{c}}$ confining. So confinement in the Higgs phase is only C confinement.

This time we have

where

Since custodial symmetry is unbroken for gauge configurations drawn from the probability distribution $P(U)$, it follows that for such configurations, in the symmetric phase at $h\rightarrow 0$,

Therefore

Note that this result holds for all $V$ operators (all isolated charges) in the symmetric phase, independent of any gauge choice. This is a consequence of the $Z_{N}$ invariance of the ground state, and it means that in the symmetric phase, unlike in the broken phase, there is a sharp distinction between charged states and color neutral states.

Expanding on this point, consider the $\mathbf{y}\rightarrow\infty$ limit in (34),

an example being $V(\mathbf{x};\infty;U)=g_{F}^{\dagger}(\mathbf{x},t;U)$, where $g_{F}(x;U)$ is the gauge transformation to an F-gauge. Then, although $\Psi_{V}$ is a physical state and therefore satisfies the Gauss Law constraint, i.e. is invariant under infinitesimal gauge transformations, it transforms covariantly under the global $Z_{N}$ subgroup of the gauge group. In other words, it is truly a charged state, and because of its transformation properties it is orthogonal to any color singlet, i.e. uncharged state. Again we stress that this is a consequence of the $Z_{N}$ symmetry of the vacuum in the symmetric state. In the broken (Higgs, spin glass) phase the vacuum is not an eigenstate of $Z_{N}$, and even if an operator acting on the vacuum has definite $Z_{N}$ transformation properties, the resulting physical state does not. In fact, this is why it is possible for $\langle\phi\rangle_{F}$ to have a non-zero expectation value in some F-gauge in the broken phase, despite the fact that the $\phi$ operator transforms non-trivially under the global $Z_{N}$ subgroup of the gauge group, and fixing to an $F$ gauge does not alter that property.

Charged states in the unbroken phase may be of either finite of infinite energy. If there are no charged states of finite energy above the vacuum energy, then the system is in an S${}_{\text{c}}$ confinement phase. If, on the other hand, there do exist charged finite energy states, orthogonal to all neutral states, then states of this kind will necessarily appear in the spectrum. The system cannot then be in a C confining phase, where there are no charged particles in the spectrum. Nor can it be in an S${}_{\text{c}}$ confining phase, where isolated charges are all states of infinite energy. The remaining possibility is a massless phase. So the phase of unbroken custodial symmetry is either S${}_{\text{c}}$ confining, or massless. This is consistent with the fact that $\langle\phi\rangle_{F}=0$ in all $F$-gauges in the symmetric phase, so there exists no sensible perturbative expansion of $\phi(x)$ around a non-zero expectation value, and no Brout-Englert-Higgs mechanism in the symmetric phase, at least not one that can be seen in any physical $F$-gauge.

The conclusion is that the spin glass phase is a C confinement (Higgs) phase, while the phase of unbroken custodial symmetry may be either a massless or an S${}_{\text{c}}$ confining phase. In the absence of a massless phase, as in SU(2) gauge Higgs theory in $D=3+1$ dimensions, the transition from the symmetric to the spin glass phase coincides with the transition from S${}_{\text{c}}$ confinement to C confinement.

It is useful to check numerically, in some examples, that

1. 1.

   The charged and neutral states defined in (34) are orthogonal in the confinement phase, and have finite overlap in the Higgs phase, as a consequence of unbroken vs. broken $Z_{N}$ symmetry;

2. 2.

   The energy of any charged state diverges with charge separation in the confinement phase, regardless of $V(\mathbf{x},\mathbf{y};U)$, but it is always possible to find operators $V(\mathbf{x},\mathbf{y};U)$ such that the resulting state has finite energy in the same limit in the Higgs phase.

This will be illustrated in SU(3) gauge Higgs theory. But before proceeding to those results, let us look at the simplest possible example of a charged state, namely pure QED with a static charge (and no dynamical charges) at the point $\mathbf{x}$. The lowest energy state of this kind was written down long ago by Dirac Dirac (1955):

where

It is easy to check that $|\Psi_{\mathbf{x}}\rangle$ satisfies the Gauss Law. However, let $g(x)=e^{i\theta(x)}$ be an arbitrary U(1) gauge transformation, and we separate out the zero mode $\theta(x)=\theta_{0}+\tilde{\theta}(x)$. Then

and therefore

So $|\Psi_{\mathbf{x}}\rangle$ transforms covariantly under the global center subgroup (which is U(1)) of the U(1) gauge group; this is the hallmark of a charged state. The vacuum state is invariant under such transformations, simply because it depends only on the gauge field, which is itself invariant under those global transformations. This means that $|\Psi_{\mathbf{x}}\rangle$ is a charged state, and of course it is finite energy (apart from the usual UV divergence which is regulated on the lattice).

The operator $\rho_{C}(\mathbf{x};A)$ is an example of what we have elsewhere Greensite and Matsuyama (2017) called a “pseudomatter” field; this is a field which transforms like a matter field in the fundamental representation, except that it is invariant under transformations in the global center subgroup of the gauge group. Any gauge transformation $g_{F}(x;U)$ to an F-gauge can be decomposed
into $N$ pseudomatter fields $\{\rho_{n}\}$, and vice-versa:

In particular, the operator $\rho_{C}^{*}(\mathbf{x};A)$ defined earlier is precisely the gauge transformation to Coulomb gauge in an abelian theory. This operator dresses a static charge with a surrounding Coulomb field. Another example, in an SU(N) lattice gauge theory, is any eigenstate $\xi_{n}(\mathbf{x};U)$ of the covariant Laplacian operator

where

These constructions underlie the Laplacian gauge introduced by Vink and Weise Vink and Wiese (1992), which is free of Gribov copies.

Now let us consider an SU(3) lattice gauge Higgs theory, with action

and we impose for simplicity a unimodular constraint $|\phi|=1$ on the Higgs field. Let us define

where

and where the $\xi_{1}(\mathbf{x};U)$ is a pseudomatter operator corresponding to the eigenstate of the lattice Laplacian operator with the smallest eigenvalue. The energy expectation values of the charged and neutral states are then given by eq. (LABEL:EV), using the operators in (50) above, and these we have computed numerically in Greensite (2020). The results obtained in the confinement phase at $\beta=5.5,\gamma=0.5$, and in the Higgs phase at $\beta=5.5,\gamma=3.5$, are shown in Figure 5. Note that in these plots, taken from Greensite (2020), the charged and neutral states are labeled

for reasons given in that reference.

These plots illustrate the statements made about the distinction between different types of confinement in the Higgs and confinement phases. The confined phase is S${}_{\text{c}}$ confining. Hence the energy of any charged state, created with any $V(\mathbf{x},\mathbf{y},U)$ operator, will diverge at quark separation $R\rightarrow\infty$. The energy of a neutral state (in this case the charge of the quarks are neutralized by the Higgs fields), will be finite in the same limit. This is all clearly seen in Figure 5(a). The overlap of charged and neutral states vanishes in the confinement phase (a prediction of eq. (39)), as we see in Figure 5(c). Things are different in the Higgs phase. In this case the energies of both the charged and neutral states go to a finite constant as $R\rightarrow\infty$ as seen in Figure 5(b), meaning that the Higgs phase is not a phase of S${}_{\text{c}}$ confinement, and in Figure 5(d) we see that the overlap between the two states is substantial (this time a prediction of eq. (35)) in the same limit, due to the fact that the the global center subgroup of the SU(3) gauge group is spontaneously broken, and there is no longer any essential distinction between charged and neutral states, so long as those states satisfy the required Gauss Law constraint.

Since our discussion has focused on gauge Higgs theory, it is natural to ask how it applies to a theory like QCD, where the matter in the fundamental representation of the gauge group is fermionic, and we would expect that the global center subgroup of the gauge group is unbroken. Although this question has not yet been addressed numerically, we believe it might be addressed in the following way: After integrating out the fermionic degrees of freedom, one ends up with a fermionic determinant, which in turn can be expressed in terms of pseudofermion fields, which are in fact scalar fields in the fundamental representation. This is how QCD can be simulated numerically, via the hybrid Monte Carlo algorithm. The pseudofermion action is, of course, non-local, but apart from computation time we see no difficulty in principle in applying the numerical approach outlined in section II.2 to QCD, where we would expect $\langle\Phi\rangle=0$ in the appropriate limit.