$0^{-+}$ pseudoscalar (the lightest $P=-$ glueball), the $E^{-+}$ where the
lightest state is the $2^{-+}$ pseudotensor, and the $T_{1}^{+-}$ which is the
lightest $C=-$ state and where the lightest 3 component state is the $C=-$
vector. We do not analyse any $R^{--}$ states since they are all too heavy.

We begin with the $T_{2}^{++}$ representation. Here our double trace operator
would project onto a two glueball state consisting of the lightest
$A_{1}^{++}$ and $T_{2}^{++}$ glueballs if there were no mixing. Neglecting
interactions its energy is just the sum of these masses. This case differs
from the $A_{1}^{++}$ representation discussed above because this is not the
lightest scattering state one can construct. Two $A_{1}^{++}$ glueballs with
two units of angular momentum are lighter, although one can expect an angular
momentum threshold suppression factor that would need to be quantified. Such a
state requires the use of operators with non-zero momentum which would mean
extending our basis of operators well beyond our choice for this exploratory
study. It is plausible that what we learn using our heavier two glueball
states would also apply to these lighter ones. So we carry out for the
$T_{2}^{++}$ representation the analogue of the analysis in Fig. 16 for the
$A_{1}^{++}$. Once again we find that the inclusion of our double trace
operators leads to an extra state although because of the denser packing of
the states it is less prominent than in the $A_{1}^{++}$ case. However the
effective energy of this state is consistent with decreasing towards that of a
scattering state composed of the lightest $A_{1}^{++}$ and $T_{2}^{++}$
glueball, albeit with substantial statistical uncertainty. So it is very
plausible that it is a scattering state composed of the lightest $A_{1}^{++}$
and $T_{2}^{++}$ glueballs. The spectrum from the double trace basis shows
quite clearly that the state that is lightest at small $t$ has an energy
roughly equal to the sum of the lightest $A_{1}^{++}$ and $T_{2}^{++}$
glueballs, showing it to be a scattering state. It is also clear that the
projection of these double trace operators onto the lightest $T_{2}^{++}$
glueball is very small. All this makes it plausible that the five or six
lightest states in the $T_{2}^{++}$ channel are single glueballs. However we
recall our above caveat that there are lighter scattering states that could,
in principle, behave differently.

With the $T_{1}^{+-}$ representation our double trace operators project onto
the lightest two glueball state since the lightest $T_{1}^{+-}$ is the
lightest $C=-$ state. Here we again find that the addition of the double trace
operators to the single trace basis does produce an extra state and that the
effective energy of that state appears to be decreasing towards the energy of
the lightest scattering state, up to the point where the errors grow too large
to allow a statement. From this we infer that the lightest five $T_{1}^{+-}$
states obtained using the single trace basis are not scattering states. In
addition we find that in the spectrum obtained using just the double trace
operators the lightest state appears to approach the expected scattering state
and that there apears to be no significant component of the lightest single
glueballs present in this basis, providing further evidence for little overlap
between our single and double trace operators.

In the case of the $A_{1}^{-+}$ representation our double trace operators
naturally project onto a state with both an $A_{1}^{++}$ and an $A_{1}^{-+}$
glueball. Just as for the $A_{1}^{++}$ this should be the lightest possible
scattering state: although one can obtain $P=-$ through unit angular momentum
one would need a glueball with non-zero spin to take us back to an overall
$A_{1}$ state. We show in Fig. 18 how the inclusion of our double trace
operators affects the spectrum. While the ground and first excited states are
unaffected, there is clearly an extra state just above these, and a
significant shift in the mass of the next state above this. The effective
energy of the (probable) extra state descends rapidly towards the energy of
the free scattering state, but the rapidly growing errors prevent us from
telling if it asymptotes to this value or continues to decrease. For further
evidence we plot in Fig. 19 the spectrum obtained in the basis with only the
double trace operators. Here we have a quite dramatic contrast to what we have
seen above in the $A_{1}^{++}$, $T_{2}^{++}$ and $T_{1}^{+-}$ representations:
the lightest state is clearly the ground state of the single trace spectrum.
This tells us that here we have a large overlap between single trace and
double trace operators. So here we can have legitimate concerns about the
presence of multiglueball states in the low-lying spectrum obtained using
single trace operators.

The $E^{-+}$ representation contains the second lightest state in the
$P,C=-,+$ sector, after the $A_{1}^{-+}$ ground state. Neglecting
interactions, our double trace operators would project onto the two glueball
states composed of an $A_{1}^{++}$ and an $E^{-+}$ glueball. There are
scattering states that are slightly lighter but they should have angular
momentum threshold factors that effectively cancel that advantage. (For
example, the lightest $A_{1}^{++}$ and $A_{1}^{-+}$ glueballs with two units
of angular momentum.) We find, when plotting the effective masses, that the
addition of the double trace operators to the single trace basis does produce
an extra state. However this state is nearly degenerate with another state so
it is unclear which is the extra one. (It is quite possible that our
variational procedure mixes the two.) In either case the effective energy of
that state appears to be decreasing towards the energy of the lightest
scattering state, up to the point where the errors grow too large to allow a
statement. The spectrum of states with just the double trace basis shows quite
clearly that the lightest state is the scattering state and that there is no
significant overlap of our double trace operators onto the low-lying spectrum
obtained with the single trace operators.

In summary, the above study in $SU(3)$ shows that two glueball states have
very little overlap onto most of the spectrum one obtains from single trace
operators for the lighter and hence most interesting states in our calculated
glueball spectra. This is something of a surprise in $SU(3)$ since one would
not expect to be able to invoke a large-$N$ suppression for $N=3$. The one
apparent exception concerns the $A_{1}^{-+}$ representation which contains the
$0^{-+}$ pseudoscalar which is of particular theoretical interest. This
clearly merits a much more detailed investigation.

We now turn to a similar study in $SU(8)$. This we should do since for $N\geq
8$ we use considerably smaller spatial volumes than for $SU(3)$, taking
advantage of the expected large-$N$ suppression of finite volume corrections.
We perform our study on a $16^{3}24$ lattice at $\beta=46.70$ which
corresponds to a lattice spacing that is slightly larger than that of the
above $SU(3)$ study if measured in units of the string tension, $\sigma$,
although equal when measured in units of the mass gap. Our results are very
much the same as for $SU(3)$: in the $A_{1}^{++},T_{2}^{++},T_{1}^{+-}$ and
$E^{-+}$ representations there appear to be very small overlaps between our
would-be two glueball operators and the single trace operators in the low-
lying part of the spectrum that is of interest to us in this paper. The
striking exception, as in the case of $SU(3)$, is the $A_{1}^{-+}$
representation. This is intriguing both because this representation contains
the interesting $J^{PC}=0^{-+}$ pseudoscalar glueball and because one would
have naively expected the large-$N$ suppression of such overlaps between
single and double trace operators to have taken effect by $N=8$. There is
perhaps some hint of this in comparing the approaches to the respective
$A_{1}^{-+}$ masses of the ground states.

These results for $SU(3)$ and $SU(8)$ provide some reassurance that our
spectra in the glueball calculations of this paper are indeed those of single
glueballs, at least for the lower lying spectrum that is of the greatest
physical and theoretical interest.

###  5.9 some comparisons

It is useful to compare our glueball results against those of earlier
calculations both as a check on the reliability of our errors, particularly
the systematic errors that are the most difficult to control. To reduce the
comparison to manageable proportions we limit ourselves to the continuum limit
of the $J^{PC}=0^{++},2^{++},0^{-+},2^{-+},1^{+-},3^{+-},2^{--}$ ground states
and to calculations that have appeared after the year 2000. The first five
states are the lightest ground states and are easy to identify, while the
sixth involves a more challenging identification of near-degeneracies amongst
the cubic irreducible representations, and the last is an example of the much
heavier states over which we have less control. We will also limit ourselves
to $SU(3)$, where there are several calculations, and to $SU(8)$ as a
representative of our larger $N$ calculations.

We list the $SU(3)$ and $SU(8)$ comparisons in Tables 39 and 40. The
calculations in [1, 2, 9] are broadly of the same type as those in the present
paper, making a comparison straightforward. The calculations of [1, 2] were
designed to explore the approach to the large $N$ limit and focussed upon the
masses of the $J^{PC}=0^{++},2^{++}$ ground states and the string tension. The
primary goal of [9] was to calculate the masses of higher spin glueballs which
cannot be identified using the naive approach of searching for near-
degeneracies, as used in the present paper, and this required the application
of novel dedicated techniques. The calculations in [10] are significantly
different to all of these, in that they use an alternative ‘improved’ lattice
action and anisotropic lattices with a temporal lattice spacing, $a_{t}$,
smaller than the spatial lattice spacing, $a_{s}$, by a factor of (roughly) 3
or 5. In addition the continuum glueball masses are expressed in units of the
Sommer scale $r_{0}$ rather than the string tension $\sigma$. We have used the
value $r_{0}\surd\sigma=1.160(6)$, as fitted in [11], to translate the values
given for $r_{o}M_{G}$ in [10] to the values $M_{G}/\surd\sigma$ listed in
Table 39.

We begin with the $SU(3)$ comparison in Table 39. Overall we see that the
results of the different calculations agree within about 2 standard deviations
for all the states. In particular the values in [9] are within one standard
deviation of our values. On the lighter ground states our errors are far
smaller than those of the earlier calculations. However on the two heaviest
states the errors in [10] are similar to ours, demonstrating the expected
advantage of highly anisotropic lattices for calculating the masses of heavy
glueballs. Focussing now on possible differences, we note that all the mass
estimates from [10] are slightly above ours, and indeed above those of [9] It
is possible that this has to do with the scale $r_{0}$. This scale is
determined as the spatial distance at which the heavy quark potential has a
certain value of its derivative. Thus its value is expressed in units of the
spatial lattice spacing, i.e. as $r_{0}/a_{s}$, and this has to be expressed
as $r_{0}/a_{t}$ in order to serve as a scale for a lattice glueball mass,
$a_{t}M_{G}$. This rescaling is clearly sensitive to even small shifts of the
anisotropy from the tree level values of 3 or 5. Moreover the values of
$a_{s}$ used in [10] are large which means that $r_{0}/a_{s}$ is not large,
and this makes the accurate determination of its value more delicate. So it is
possible that all this is enough to lead to a $1-2\%$ shift in the estimates
of $M/\surd\sigma$ in this calculation. Turning now to [1, 2] we see that the
$0^{++}$ mass is about two standard deviations higher than our value (or that
of [9] ). A possible source of this is the fact that these older calculations
have a more limited range of higher $\beta$ values and include in their
continuum extrapolations the smaller values of $\beta$ where there is some
effect of the dip in the mass gap associated with the strong-to-weak coupling
bulk transition, as discussed in Section 5.4. This dip in the mass at larger
lattice spacings may then bias the straight line fit in $a^{2}\sigma$ so that
its $a=0$ intercept is a little too high. In the present calculations the
statistical errors are small enough that a good straight line fit is no longer
possible if one includes such large $a$ values and by excluding them any such
bias becomes smaller and the continuum intercept is a little lower. This
discussion is only relevant to the mass gap, since the other glueballs appear
to be insensitive to the bulk transition and have, for the most part, a small
dependence on the size of the lattice spacing.

Turning now to the $SU(8)$ comparison in Table 40, we see that for the states
other than the $0^{++}$ the masses from the different calculations agree
within one standard deviation, although the errors on the earlier calculations
are typically about ten times larger than ours, leaving room for potential
discrepancies. The mass gap shows a signficant discrepancy but both of the
earlier calculations include masses calculated close to the bulk transition at
$\beta=\beta_{b}^{\downarrow}$ where the mass gap appears to have a
significant dip, as discussed in Section 5.4, which may well explain this
minor difference.

The above comparisons naturally raise the question of why we chose to use the
traditional Wilson plaquette action instead of some alternative ‘improved’
action, such as the one used in [10]. In glueball calculations the main
improvement observed, for $SU(3)$, is to largely remove the dip in the mass
gap as one transitions from strong to weak coupling. This means that
calculations at coarser lattice spacings, and so on smaller lattices, may be
used in the continuum extrapolation of the mass gap. However a larger lattice
spacing means that the correlator signals tends to disappear into the
statistical noise before one has a well defined effective mass plateau. To
remedy this one can use an anisotropic lattice action with the temporal
spacing $a_{t}$ much less than the spatial $a_{s}$. However using contemporary
resources there is no need to worry about the dip in the mass gap: one can
easily do calculations far from the dip, at much weaker coupling, as we have
seen in this paper. Moreover if one works at coarse lattice spacings using
such an improved action one does not know in advance whether the behaviour of
other glueballs will be improved or worsened. And in any case, as we have seen
in Figs 10,11,12, the lattice corrections to the glueball masses are already
quite modest with the standard Wilson plaquette action. An advantage of the
simple plaquette action is that it is known to have exact reflection
positivity and hence the transfer matrix has real positive eigenvalues between
zero and one, just as if it was the exponential of a Hamiltonian. This means
that one does not need to be cautious in applying a variational calculation of
the glueball spectrum. Of course one can use an anisotropic plaquette action,
which will also possess reflection positivity. Such a calculation would
produce a finer resolution in time of the correlator and so could be very
useful for calculating heavier glueball masses. However it has a disadvantage
in setting the scale as described just above. One needs to know the relation
between $a_{s}$ and $a_{t}$ and since the chosen tree level anisotropy will
receive corrections, this need to be estimated, and the error on this estimate
may be significant. Thus our suggestion for a better calculation than the
present one is that one does two calculations in parallel. One calculation
should be with an isotropic action, and this will provide very accurate
continuum values for the light glueballs, including the mass gap $M_{G}$, in
units of, for example, the string tension, $\sigma$, or the Sommer scale
$r_{0}$. The second calculation should be with a strongly anisotropic action
and provided that one has a large enough basis of operators, this should
provide accurate calculations of the heavier glueball masses in units of, say,
the mass gap. Here it is important that the spatial lattice spacings should
extend to small values, just as in the isotropic calculations, in order to
guarantee the credibility of the continuum extrapolations. Putting together
both calculations allows all the masses to be expressed in units of $\sigma$
or $r_{0}$. The action used could be either the plaquette action or some
improved action, although maintaining reflection positivity is desirable from
a practical point of view.

##  6 Topological fluctuations

Euclidean $D=4$ $SU(N)$ gauge fields possess non-trivial topological
properties, characterised by a topological charge $Q$ which is integer-valued
in a space-time volume with periodic boundary conditions. This charge can be
expressed as the integral over Euclidean space-time of a topological charge
density, $Q(x)$, where

$Q(x)=\frac{1}{32\pi^{2}}\epsilon_{\mu\nu\rho\sigma}\mathrm{Tr}\\{F_{\mu\nu}(x)F_{\rho\sigma}(x)\\}.$
(30)

Since the plaquette matrix $U_{\mu\nu}(x)=1+a^{2}F_{\mu\nu}(x)+....$ on
sufficiently smooth fields, one can write a lattice topological charge density
$Q_{L}(x)$ on such fields as

$Q_{L}(x)\equiv{\frac{1}{32\pi^{2}}}\epsilon_{\mu\nu\rho\sigma}\mathrm{Tr}\\{U_{\mu\nu}(x)U_{\rho\sigma}(x)\\}=a^{4}Q(x)+O(a^{6}).$
(31)

However this definition lacks the reflection antisymmetry of the continuum
operator in eqn(30), since all the plaquettes $U_{\mu\nu}(x)$ are defined as
forward going in terms of our coordinate basis, so to recover these symmetry
properties we use the version of this operator that is antisymmetrised with
respect to forward and backward directions [55] (although this is unnecessary
on smooth fields).

The fluctuations of $Q_{L}(x)$ are related to the expectation value of the
composite operator $Q_{L}^{2}(x)$ whose operator product expansion contains
the unit operator [55], so these fluctuations are powerlike in $1/\beta$. On
the other hand, the average of $Q_{L}(x)$ is $O(a^{4})$ and hence
exponentially suppressed in $\beta$. Thus as $\beta$ increases the
fluctuations around $Q_{L}(x)$ and $Q_{L}=\sum_{x}Q_{L}(x)$ diverge compared
to the physically interesting mean values. In addition the composite operator
$Q_{L}(x)$ also receives a multiplicative renormalisation $Z(\beta)$ such that
$Z(\beta)\ll 1$ at accessible values of $\beta$ [56].

In practice all this means that one cannot extract the topological charge of a
typical lattice gauge field by directly calculating $Q_{L}=\sum_{x}Q_{L}(x)$
on that gauge field. However we note that the fluctuations obscuring the value
of $Q$ are ultraviolet, while the physically relevant topological charge is on
physical length scales. Thus if we perform a very limited local smoothening of
the fields to suppress the ultraviolet fluctuations, this should not affect
physics on long distance scales, and the value of $Q_{L}=\sum_{x}Q_{L}(x)$
calculated on these smoothened fields should provide a reliable estimate of
$Q$. Moreover, recalling that the total topological charge of a gauge field is
unchanged under smooth deformations, we can expect that even under a
moderately large amount of continued smoothening the value of $Q_{L}$ will not
change, even though $Q_{L}(x)$ itself does gradually change. One convenient
way to smoothen the gauge fields is to locally minimise the action. Such a
‘cooling’ of the original ‘hot’ lattice gauge field [57] involves sweeping
through the lattice one link at a time, precisely like the Monte Carlo except
that one chooses the new link matrix to be the one that minimises the total
action of the plaquettes containing that link matrix. This is a standard
technique that one can find described in more detail in, for example, [58]. An
alternative and attractive smoothing method with perturbatively proven
renormalisation properties is the gradient flow [59, 60, 61] which has been
shown to be numerically equivalent to cooling [62, 63, 64]. Cooling typically
performs nearly two orders of magnitude faster than the gradient flow and
since we are aiming for large statistics we adopt the simplest cooling method.
After the first couple of cooling sweeps the fields are already quite smooth,
as we shall see below. Since we are minimising the action and since in the
continuum the minimum action field with a given $Q$ is a multi-instanton
field, we expect that under systematic cooling the lattice field will be
driven to become some multi-instanton field, which one can see by calculating
the distribution $Q_{L}(x)$ on such a field. Of course, because of the
discretisation of space-time the topological properties of lattice fields are
only approximately like those of a continuum field. One can deform a large
instanton by gradually shrinking its non-trivial core and on a lattice this
core can shrink to within a hypercube. At this point what was an instanton has
been transformed into a gauge singularity and the value of $Q$ will now differ
from its original value by one unit. (Equally, one can gradually grow an
instanton out of a hypercube.) This process can occur during cooling but it
can equally occur during the course of our Monte Carlo. In the latter case, it
is these changes in $Q$ that allow us to sample all possible values of $Q$ and
hence maintain the ergodicity in $Q$ of our Markov process. As $\beta\uparrow$
the distance between physical and ultraviolet scales grows and these changes
in $Q$ become increasingly suppressed – in fact more strongly than the usual
critical slowing down (see below). This is as it should be: after all, the
changes in topological charge are no more than a (useful) lattice artifact.

We have outlined in Section 2.4 the specific reasons for the suppression of
changes in $Q_{L}$ when $a(\beta)\to 0$ at fixed $N$ and when $N\to\infty$ at
fixed $a(\beta)$, and how we deal with this problem in our glueball and string
tension calculations. Below, in Section 6.1, we will give some evidence that
the value of $Q_{L}$ on a cooled lattice field does indeed reflect the
topology of the original lattice field. Then, in Section 6.2, we will give
some results on the rate of critical slowing down, both as $a(\beta)\to 0$ and
as $N\to\infty$. In Section 6.3 we present our results for the continuum
topological susceptibility in those cases, $2\leq N\leq 6$, where the
topological freezing is not too serious an obstacle. Finally in Section 6.4 we
present our numerical results for the factor $Z(\beta)$ that relates the value
of $Q_{L}$ before cooling to its value, $Q_{I}$, after cooling. Our study of
lattice topology has some limitations of course: it is not a dedicated study
but is constrained by being a byproduct of our glueball and string tension
calculations. Finally we remark that it is only in the calculations of the
topological susceptibility that the topological freezing is an obstacle; in
our other calculations in this Section it does not matter whether the
topological charge we study is introduced by hand or appears spontaneously.

###  6.1 topology and cooling

As we cool our lattice fields the fluctuations in the measured value of
$Q_{L}$ decrease and it rapidly settles down to a value that is close to an
integer. As a typical example we plot in Fig.20 the number of fields with a
given topological charge $Q_{L}$ after 2 coolings sweeps and the number after
20 cooling sweeps, taken from sequences of $SU(5)$ fields on a $16^{3}20$
lattice at $\beta=17.63$. With respect to the calculations in this paper, this
corresponds to an intermediate value of $N$, and of $a(\beta)$, and of the
lattice volume. As we see, after 20 cooling sweeps the values of $Q_{L}$ are
very strongly peaked close to integer values. The reason that the peak is not
at an integer, even when the cooling has erased the high frequency
fluctuations, is that for an instanton of size $\rho$ on a lattice, we only
obtain $Q_{L}\to 1$ in the limit $\rho/a\to\infty$, with deviations from unity
that are powers of $a/\rho$. However, as is clear from Fig.20, there is no
significant ambiguity in assigning to each field after 20 cooling sweeps an
integer valued topological charge which we label $Q_{I}$. We can expect
$Q_{I}$ to provide our most reliable estimate of the topological charge $Q$ of
the original lattice gauge field. For the calculations with the largest values
of $a(\beta)$ identifying the value $Q_{I}$ can be less clear-cut, but such
lattices are of little importance in determining continuum physics. We also
see from Fig.20 that even after only 2 cooling sweeps the fields fall into
groups that only overlap slightly. Since 2 sweeps cannot affect anything other
than the most local fluctuations, we can assume that this segregation into
differing topological sectors directly reflects the topology of the gauge
fields prior to any cooling.

As a second example we show in Fig.21 the same type of plot for several
sequences of $SU(8)$ fields generated at $\beta=47.75$ on a $20^{3}30$
lattice. This corresponds to our smallest lattice spacing in $SU(8)$. Here we
see an even sharper peaking after 20 cooling sweeps and, more interestingly, a
very clear separation between the various sectors even after only 2 cooling
sweeps. In contrast to the $SU(5)$ example, in this case there is essentially
no tunnelling between topological sectors, and the observed distribution of
$Q$ has been imposed on the starting configurations for the various sequences.

The loss of ergodicity with respect to the topological charge is illustrated
for $SU(8)$ in Fig.22 where we show the values of $Q_{L}$ after 2 and 20
cooling sweeps taken every 100 Monte Carlo sweeps for two sequences of 50000
sweeps generated at $\beta=47.75$. In one sequence we have $Q_{L}\sim 1$ and
in the other we have $Q_{L}\sim 2$. It is interesting that even after only 2
cooling sweeps the separation in the measured values of $Q_{L}$ is
unambiguous. A similar plot for $SU(5)$ fields generated at $\beta=17.63$ is
shown in Fig.23. Here the value of $Q_{L}$ remains unchanged over subsequences
that are typically a few thousand sweeps long, but the changes are
sufficiently frequent that in our overall ensemble of $\sim 2\times 10^{6}$
fields we may regard $Q_{L}$ as ergodic. But just as for $SU(8)$ the values of
$Q_{L}$ after only 2 cooling sweeps track the values after 20 cooling sweeps.

The close relationship that we observe in Fig.22 and Fig.23 between the values
of $Q_{L}$ after 2 and 20 cooling sweeps is confirmed explicitly in Fig.24.
Here we have taken all the $\beta=47.75$ $SU(8)$ fields on which we have
calculated $Q_{L}$ and we have extracted the three subsets that correspond to
$Q_{L}\simeq 1,2,3$ after 20 cooling sweeps. For each subset we plot the
values that $Q_{L}$ takes on fields after 2 cooling sweeps. As we see in
Fig.24 the three distributions do not overlap. So we can assign to each field
a value of $Q$ that is completely unambiguous (at least for our statistics) on
the basis of the value of $Q_{L}$ measured after only 2 cooling sweeps, where
the long-distance physics should be essentially unchanged from that of the
original uncooled lattice fields. As an aside, we show in Fig.25 the same plot
as in Fig.24, but with values of $Q_{L}$ calculated after only 1 cooling
sweep. Here the distributions are broader, so that one can no longer use the
value of $Q_{L}$ to unambiguously assign the field its value of $Q$, but the
fact that even after 1 cooling sweep the values of $Q_{L}$ strongly reflect
the value of $Q$ after 20 cooling sweeps is evident.

It is instructive to see how the measured values of $Q_{L}$ vary with the
number of cooling sweeps $n_{c}$. As an example, we take an ensemble of fields
that have a topological charge $Q=2$, as determined by the value of $Q_{L}$
after 20 cooling sweeps. For this same ensemble we calculate how the average
value of $Q_{L}$ varies with $n_{c}$ and also how the standard deviation of
its fluctuations vary with $n_{c}$. These quantities, labelled
$\overline{Q}_{L}$ and $\sigma_{Q_{L}}$ respectively, are shown in Table 41
for three different values of $\beta$ in $SU(8)$. As we see, in the uncooled
($n_{c}=0$) fields the fluctuations $\sigma_{Q_{L}}$ are so large compared to
the average charge $\bar{Q}_{L}(n_{c}=0)$ of those fields, that one cannot
hope to estimate for individual fields the true value of $Q$ from the value of
$Q_{L}$ at $n_{c}=0$. However we also see that even after only 1 cooling sweep
the value of $\bar{Q}_{L}/\sigma_{Q_{L}}$ increases by a factor $\sim 20$ and
after 2 cooling sweeps by $\sim 80$, so that one can, with rapidly increasing
reliability, assign a value of $Q$ to an individual field on the basis of the
value of $Q_{L}(n_{c})$ at the first few cooling sweeps. What we also see in
Table 41 is that beyond the lowest few values of $n_{c}$ there is little
difference between the 3 ensembles, despite the fact that the lattice spacing
changes by a factor $\sim 1.7$. For larger $n_{c}$ the value of $Q_{L}(n_{c})$
decreases slightly with decreasing $\beta$ as one expects because the typical
instanton size will decrease in lattice units and the discretisation
corrections are negative. For $n_{c}=0$ the decrease in $Q_{L}$ is larger and
reflects the $\beta$-dependence of the renormalisation factor $Z(\beta)$ which
is driven by high frequency fluctuations, as analysed more quantitatively in
Section 6.4. The rapid increase of $\sigma_{Q_{L}}$ with increasing $\beta$ is
driven by two competing factors: the high frequency fluctuations per lattice
site decrease as an inverse power of $\beta$, but since these fluctuations are
roughly uncorrelated across lattice sites, there is a factor proportional to
the square root of the lattice volume in lattice units which increases
exponentially with $\beta$ if our volumes are roughly constant in physical
units. At larger $\beta$ the latter factor will dominate and it therefore
becomes rapidly harder to relate $Q$ from $Q_{L}(n_{c}=0)$ as we approach the
continuum limit. On the other hand, after the first few cooling sweeps the
high frequency fluctuations have been largely erased and the fluctuations of
$Q_{L}$ no longer increase with increasing $\beta$, so that it becomes easier
to identify the value of $Q$ from the value of $Q_{L}$.

As a footnote to the above we have calculated the same quantities as in Table
41 but now for different $SU(N)$ groups. The calculations are at roughly the
same value of $a\surd\sigma$ as for the $SU(8)$ fields at $\beta=46.70$ in
Table 41, i.e. at roughly the same value of the ’t Hooft coupling, so that the
contribution of the high frequency fluctuations is roughly the same. We find
that the variation in $\sigma_{Q_{L}}(n_{c}=0)$ with $N$ is consistent with
being due to the difference in the square root of the lattice volumes. At
larger $n_{c}$ the value of $\sigma_{Q_{L}}(n_{c})$ decreases with increasing
$N$, which tells us that larger-scale non-perturbative fluctuations, such as
in the instanton size, are decreasing. Finally, the main practical point is
that there is very little variation with $N$ of $\overline{Q}_{L}$ at any
value of $n_{c}$.

###  6.2 tunneling between topological sectors

In the continuum theory $Q$ cannot change under continuous deformations of the
fields unlike most other quantities, such as glueball correlation functions,
so one expects to lose ergodicity in $Q$ much faster than in such other
quantities as one approaches the continuum limit in a lattice calculation
using a local Monte Carlo algorithm such as the heat bath. In this section we
shall provide our data on this ‘freezing’ of $Q$ as a function of $a$ and $N$
and then compare this to some theoretical expectations.

As described earlier, the value of $Q$ changes if the core of an instanton
shrinks and disappears within a hypercube (or the reverse of this process).
When $a$ is small enough such instantons can be described by dilute gas
calculations since these very small instantons are very rare and the effect on
them of other background field fluctuations that exist on physical length
scales will be negligible. So the basic process is for $Q$ to change by one
unit. Therefore we use as our measure of topological freezing the average
number of sweeps between changes of $Q$ by one unit. We call this $\tau_{Q}$.
Since the probability of such a process is clearly proportional to the space-
time volume, which is not something we have tried to keep constant in our
calculations, we rescale our measured values of $\tau_{Q}$ to a standard
physical volume, which we choose to be $V_{0}=l^{4}$ with $l\surd\sigma=3.0$,
so that we can compare the results of different calculations. Our calculations
of $Q$ have not been performed after every sweep, but typically with gaps of
25 or 50 or 100 sweeps depending on the calculation. Our estimates of
$\tau_{Q}$ can only be reliable if they are much larger than this gap, since
otherwise there could be multiple changes of $Q$ within the gaps that we are
missing, and so we do not includes those ensembles where this issue arises –
typically at the coarser lattice spacings. However even if $\tau_{Q}$ appears
to be much larger than the gap, we will occasionally see that $\Delta Q$, the
change in $Q$ across a gap, is greater than unity. In this case we assume that
there have been $|\Delta Q|$ jumps within the gap and we make that part of our
final estimate of $\tau_{Q}$. In practice this makes a very small difference
to our results. By counting the number of changes of $Q$ in our total sequence
of lattice fields we can obtain the average distance between such changes,
$\tau_{Q}$, once we have renormalised to our standard volume $V_{0}$.

As $\tau_{Q}\to\infty$ the above definition is adequate. However when
$\tau_{Q}$ is not very large one can worry about the unwanted contribution of
near-dislocations that occur across a measurement of $Q$. We have in mind a
small instanton that appears out of a hypercube shortly before a measurement,
survives the cooling (because of its environment) and so contributes to the
value of $Q$, but then quickly disappears without becoming a larger physical
instanton long before the next measurement. In a sequence of measurements of
$Q$ such an event would be characterised by a jump $\Delta Q=\pm 1$ at one
measurement followed by the opposite jump, $\Delta Q=\mp 1$, at the next
measurement. Of course it could be that when this happens we are seeing two
independent events, with an instanton appearing from one hypercube and after
the measurement an anti-instanton appearing out of a quite different
hypercube. The characteristic of the latter events is that the signs of the
changes in $Q$ at neighbouring measurements are uncorrelated. Correcting for
these we obtain the measure $\tilde{\tau}_{Q}$ (normalised to our standard
volume) which excludes this estimate of near-dislocations. Whether this
estimate is entirely reliable is arguable, so it is reassuring that the
differences between $\tilde{\tau}_{Q}$ and $\tau_{Q}$ are not large, and which
one we use does not alter our conclusions below.

We present in Table 42 our results for $\tau_{Q}$ and $\tilde{\tau}_{Q}$ from
the sequences of fields generated in our glueball and string tension
calculations. The excluded values of $\beta$ either correspond to cases where
$\tau_{Q}$ is not much larger than the gap between measurements, and this
includes all of our $SU(2)$ calculations, or where $\tau_{Q}$ has become so
large that we see no changes in $Q$ at all, which includes almost all of our
$SU(10)$ and $SU(12)$ calculations. The prominent qualitative features of our
results are that for any given gauge group both $\tau_{Q}$ and
$\tilde{\tau}_{Q}$ increase rapidly as $a(\beta)$ decreases and, at roughly
equal values of $a(\beta)$, they increase rapidly as $N$ increases.

One can formulate some theoretical expectations for the behaviour of
$\tau_{Q}$ as one decreases $a$ and increases $N$. Just before shrinking
through a hypercube an instanton will be very small with size $\rho\sim a$
where the number density, $D(\rho)d\rho$ can be estimated using the standard
semiclassical formula [21]

$D(\rho)\propto\frac{1}{\rho^{5}}\frac{1}{g^{4N}}\exp\left\\{-\frac{8\pi^{2}}{g^{2}(\rho)}\right\\}\stackrel{{\scriptstyle
N\to\infty}}{{\propto}}\frac{1}{\rho^{5}}\left\\{\exp\left\\{-\frac{8\pi^{2}}{g^{2}(\rho)N}\right\\}\right\\}^{N}\stackrel{{\scriptstyle\rho=a}}{{\propto}}\left(a\Lambda\right)^{\frac{11N}{3}-5},$
(32)

where $\Lambda$ is the dynamical length scale of the theory. This is of course
a very asymptotic expression: we have neglected the powers of $g$ because they
only contribute powers of $\log(a\Lambda)$, and we have used the 1-loop
expression for $g^{2}(a)$ which is, as we have seen in Section 4, inadequate
for our range of bare couplings. We also note that this expression only tells
us what is the probability of a very small instanton. In addition there will
be a factor for the small instanton with $\rho\sim O(a)$ to finally shrink
completely within a hypercube: this ‘tunneling event’ may contribute an
important factor depending on the lattice action being used.

The first qualitative feature of eqn(32) is that if we increase $N$ at fixed
$a$, we should find

$\tau_{Q}\propto\frac{1}{D(\rho)}\propto\left\\{\frac{1}{a\Lambda}\right\\}^{\frac{11N}{3}-5}\longrightarrow\ln\\{\tau_{Q}\\}=b+cN,$
(33)

where $c$ depends on the value of $a$ and $b$ is some undetermined constant.
In our simulations we have some that correspond to almost equal values of
$a\surd\sigma$, and hence of $a\Lambda$, across several values of $N$. These
are for $N=3,4,5,6$ at $a\surd\sigma\simeq 0.15$ and for $N=8,10,12$ at
$a\surd\sigma\simeq 0.33$. In Fig. 26 we plot the values of
$\ln\\{\tilde{\tau}_{Q}\\}$ against $N$ and we see that the behaviour is
roughly linear as predicted from eqns(32,33).

The second qualitative feature of eqn(33) is that if we vary $a$ at fixed $N$,
we should find

$\tau_{Q}\propto\frac{1}{D(\rho)}\propto\left\\{\frac{1}{a\Lambda}\right\\}^{\frac{11N}{3}-5}\longrightarrow\ln\\{\tau_{Q}\\}=b+\left\\{\frac{11N}{3}-5\right\\}\ln\left\\{\frac{1}{a\Lambda}\right\\}.$
(34)

In Fig. 27 we show plots of $\ln\\{\tau_{Q}\\}$ versus
$\ln\\{1/a\surd\sigma\\}$ for $N\in[3,8]$ and we see that the plots are
roughly linear as predicted by eqns(32,34). (As an aside, the fact that the
$SU(2)$ values of $\tau_{Q}$ never become large enough to be useful is no
surprise given that the asymptotic dependence predicted by eqn(34) is quite
weak, $\tau_{Q}\propto a^{7/3}$.) The fitted coefficients of the
$\ln\\{1/a\surd\sigma\\}$ term are listed in Table 43 and compared to the
value in eqn(34). We do not expect a good agreement since we know that the one
loop expression for $g^{2}(a)$ is a poor approximation in our range of bare
couplings but it is interesting that our calculated values listed in Table 43
do reflect the trend of the asymptotic theoretical values.

###  6.3 topological susceptibility

The simplest topological quantity that one can calculate is the topological
susceptibility, $\chi_{t}=\langle Q^{2}\rangle/V$, where $V$ is the space-time
volume. This is a particularly interesting quantity because of the way it
enters into estimates of the physical $\eta^{\prime}$ mass through the Witten-
Veneziano sum rule [66, 67]. There have been many calculations of this
quantity and here we will add to these our calculations for the gauge groups
$SU(2)$, $SU(3)$, $SU(4)$, $SU(5)$ and for those of our $SU(6)$ ensembles
where there are enough fluctuations in the value of $Q$ to make an estimate
plausible.

We typically calculate the topological charge $Q$ after every 25 or 50 or 100
Monte Carlo sweeps on most of the lattice ensembles that we use for our
glueball and/or string tension calculations. We estimate the value of $Q$ from
the value of the lattice $Q_{L}$ after 20 cooling sweeps. The assignment of an
integer value, $Q_{I}$, after 20 cooling sweeps is, as we have seen, largely
unambiguous. In Tables 44, 45 and 46 we list our values for $Q^{2}$ on the
lattices and at the couplings shown.

To obtain the continuum limit we perform a conventional extrapolation

$\left.\frac{\chi_{t}^{\frac{1}{4}}}{\surd\sigma}\right|_{a}=\left.\frac{\chi_{t}^{\frac{1}{4}}}{\surd\sigma}\right|_{0}+ca^{2}\sigma,$
(35)

where we systematically remove from the fit the values corresponding to the
largest $a$ until we get an acceptable fit. We show the resulting continuum
extrapolations for $N=2,3,4,5$ in Fig.28 and list the resulting continuum
values in Table 47. For $SU(2)$ and $SU(3)$ we do not include the value at the
smallest $a$ since it is clearly too low and it is plausible that it is due to
a gradual loss of ergodicity in $Q$ accompanied by an increasingly unreliable
estimate of the errors. We see from Fig.28 that As we increase $N$ our fits
are able to include values from increasingly coarse $a(\beta)$. This is
presumably related to the fact that the ‘bulk’ cross-over/transition between
strong and weak coupling becomes rapidly sharper as $N$ increases [15]. (It is
particularly smooth for $SU(2)$.) As shown in Table 47 all the fits are
acceptable. We have performed separate continuum extrapolations of the
susceptibilities obtained from the non-integer lattice charges, $Q_{L}$, and
the integer charges, $Q_{I}$, labelling these $\chi_{L}$ and $\chi_{I}$
respectively. In the continuum limit these should be the same and we see from
Table 47 that this is indeed so, within errors, for all except the case of
$SU(2)$, where the difference between the two values can be taken as a
systematic error that is additional to the statistical errors.

Once we have the continuum susceptibilites we can extrapolate them to
$N=\infty$ as shown in Fig.29:

$\left.\frac{\chi_{I}^{\frac{1}{4}}}{\surd\sigma}\right|_{N}=0.3681(28)+\frac{0.471(15)}{N^{2}},$
(36)

and

$\left.\frac{\chi_{L}^{\frac{1}{4}}}{\surd\sigma}\right|_{N}=0.3655(27)+\frac{0.448(15)}{N^{2}}.$
(37)

The two are within errors as one would expect. We note that this value is
consistent within errors with the $N=\infty$ extrapolation in [68] which uses
a novel technique [69] for ameliorating the problem of topological freezing.

Finally we return to the case of $SU(3)$ since it also has some
phenomenological interest. Here our analysis differs slightly from our earlier
analysis in [11] and our final result

$\left.\frac{\chi_{I}^{\frac{1}{4}}}{\surd\sigma}\right|_{SU(3)}=0.4246(36)$
(38)

is about one standard deviation higher. To transform this into physical units
we translate from units in terms of $\surd\sigma$ to the standard scale
$r_{0}$ and then to $\mathrm{MeV}$ units just as we did in eqn(22) of Section
4.2, giving

$r_{0}\surd\sigma=1.160(6)\Longrightarrow\chi_{I}^{\frac{1}{4}}=206(4)\mathrm{MeV}.$
(39)

This is within errors of the value we presented in [11] and the value
$\chi_{I}^{\frac{1}{4}}\stackrel{{\scriptstyle SU3}}{{=}}208(6)\mathrm{MeV}$
obtained in [70] using the gradient flow technique. As an aside we note that
the value of $\chi_{I}^{\frac{1}{4}}/\surd\sigma$ at $N=\infty$, as given in
eqn(36), is $\sim 13\%$ lower than the $SU(3)$ value, i.e. $\sim 179$‘MeV’ if
we simply rescale the value in eqn(39).

###  6.4 ${Z_{Q}(\beta)}$ and lattice $\theta$ parameter

Consider the ensemble of Monte Carlo lattice gauge fields that correspond to
an integer valued topological charge $Q$. If we calculate the charge $Q_{L}$
of each of these fields, prior to any cooling, the average value will be
related to $Q$ by [56]

$<Q_{L}>=Z_{Q}(\beta)Q,$ (40)

where $Z_{Q}(\beta)$ will depend (weakly) on $N$ and negligibly on the lattice
volume (as long as it is not very small). Since the deviation of
$Z_{Q}(\beta)$ from unity is driven by high frequency lattice fluctuations, it
is of little physical interest in itself. However its value is important if,
for example, one wishes to study the $\theta$ dependence of the theory by
adding a term $i\theta Q$ to the continuum action and, correspondingly, a term
$i\theta_{L}Q_{L}$ to the lattice action, since one sees that [71]

$\theta_{L}\simeq Z^{-1}_{Q}(\beta)\theta.$ (41)

Primarily for this reason we have calculated $Z_{Q}(\beta)$ in our lattice
calculations and will also provide interpolating functions that will give
$Z_{Q}(\beta)$ at values of $\beta$ that lie between our measured values.

A first estimate for $Z_{Q}(\beta)$ can be obtained in perturbation theory,
giving at one loop [56]

$Z_{Q}(\beta)\stackrel{{\scriptstyle\beta\to\infty}}{{=}}1-(0.6612N^{2}-0.5)\frac{1}{\beta}+O(\beta^{-2})$
(42)

which already tells us that for our range of $\beta$ we will have
$Z_{Q}(\beta)\ll 1$. The fact that the one loop correction is so large tells
us that the one loop estimate is likely to be not very accurate, and indeed
that proves to be the case. Moreover as we increase the lattice spacing the
typical instanton becomes smaller and the value of $Q_{L}$ acquires
significant corrections that are powers of $a(\beta)$ and which are additional
to any perturbative corrections. So in constructing our interpolating function
for $Z_{Q}$ we simply use the form

$Z^{int}_{Q}(\beta)=1-z_{0}g^{2}N-z_{1}(g^{2}N)^{2}\quad;\quad
g^{2}N=\frac{2N^{2}}{\beta},$ (43)

where $g^{2}N$ is the ’t Hooft coupling, and we make no attempt to constrain
the parameters $z_{0}$ and $z_{1}$ to perturbative values. This turns out to
be an adequate fitting function to our values of $Z_{Q}(\beta)$. It is however
important to note that while this works as an interpolating function, it is
likely to fail increasingly badly the further one uses it away from the fitted
range of $\beta$ as an extrapolating function.

Our values for $Z_{Q}(\beta)$ are obtained from fits such as those shown in
Fig.30, and are listed in Tables 48 and 49. Interpolating these values with
eqn(43) gives the values for $z_{0}$ and $z_{1}$ listed in Table 50.
Physically the most relevant interpolating function is the one for $SU(3)$:

$Z^{int}_{Q}(\beta)\stackrel{{\scriptstyle
su3}}{{=}}1-0.162(10)g^{2}N-0.0425(31)(g^{2}N)^{2}\quad,\quad\chi^{2}/n_{df}=0.62$
(44)

In the case of $SU(2)$ we have only a few entries because most of our earliest
calculations did not include calculating $Q_{L}$ on the fields prior to
cooling. In addition in the case of $SU(2)$ the identification of an integer
$Q$ after 20 cooling sweeps possesses small but significant ambiguities, which
rapidly disappear as $N$ increases. We can also fit our interpolating
functions in $N$, thus obtaining

$z_{0}=0.179(12)-\frac{0.08(15)}{N^{2}}\quad,\quad\chi^{2}/n_{df}=1.00$ (45)

and

$z_{1}=0.0482(46)-\frac{0.072(50)}{N^{2}}\quad,\quad\chi^{2}/n_{df}=1.13$ (46)

which should be reliable over a wide range of $N$ as long as we are not too
far outside the range of the t’Hooft coupling $\lambda=g^{2}N$ of our
calculations in Tables 48 and 49. Finally we remind the reader that all these
results for $Z_{Q}(\beta)$ only apply to calculations with the standard Wilson
plaquette action and with the definition of the lattice topological charge
$Q_{L}$ used here.

##  7 Conclusions

Our primary goal in this paper has been to calculate the glueball spectra of a
range of $SU(N)$ gauge theories, in the continuum limit, with enough precision
to obtain plausible extrapolations to the theoretically interesting $N=\infty$
limit. This provides the first calculation of the masses of the ground states
in all the $R^{PC}$ channels, as well as some excited states in most channels,
in the continuum limit of the $N\to\infty$ gauge theory.

Our results, for $N=2,3,4,5,6,8,10,12$, were obtained using standard lattice
gauge theory methods. Although the issue of topological freezing at larger $N$
in $SU(N)$ gauge theories is not expected to be important for glueball spectra
[28, 29], we confirmed this explicitly in some extensive $SU(8)$ calculations,
and in addition we chose to minimise any remnant bias at larger $N$ by
modifying the usual update algorithm, explicitly imposing the expected
distribution of topological charge on the starting lattice fields of our
ensemble of Monte Carlo sequences. We employed a large basis of single-trace
glueball operators, which allowed us to calculate the ground state and some
excited states for each of the $R^{PC}$ channels, where $R$ labels the
representation of the rotation symmetry group appropriate to our cubic
lattice, and $P,C$ label the parity and charge conjugation. The large basis
gives us confidence that we are not missing any low-lying states and this in
turn allows us to match near-degeneracies between states with different $R$ so
as to assign continuum spin quantum numbers to a significant number of
glueball states.

Our results have greatly extended existing calculations while largely
confirming existing results; in particular the important conclusion that
$SU(3)$ is ‘close to’ $SU(\infty)$. As before, one finds that the lightest
glueball is the $J^{PC}=0^{++}$ scalar ground state, with a mass that ranges
from $M_{0^{++}}\sim 3.41\surd\sigma$ for $SU(3)$ to $M_{0^{++}}\sim
3.07\surd\sigma$ for $SU(\infty)$, where $\sigma$ is the string tension, and
that the next heavier glueballs are the tensor with a mass $M_{2^{++}}\simeq
1.5M_{0^{++}}$, and the pseudoscalar, $0^{-+}$, which is nearly degenerate
with the tensor. One then has the $1^{+-}$ with $M_{1^{+-}}\sim
1.85M_{0^{++}}$, and this is the only relatively light $C=-$ state. At roughly
the same mass is the first excited $0^{++}$ and then the lightest pesudotensor
with $M_{2^{-+}}\sim 1.95M_{0^{++}}$. All other states are heavier than twice
the lightest scalar, with most of the $C=-$ ground states being very much
heavier than that. One sees a number of near-degerenacies which may or may not
be significant. The continuum glueball masses (in units of the string tension)
for the various $R^{PC}$ channels are listed in Tables 23-31 and for the
$J^{PC}$ channels in Tables 34-38.

Since our calculations are on a finite spatial volume we have had to identify
and exclude the ‘ditorelon’ states composed of a pair of mutually conjugate
flux tubes that wind around our periodic spatial torus. These are states whose
projection onto our single trace operators will vanish as $N\to\infty$. We
also need to exclude any multiglueball scattering states. Our (albeit partial)
explicit calculations using the corresponding double trace operators strongly
suggest that these states do not appear in the glueball spectra that we claim
to obtain using our single trace basis.

In calculating the glueball spectra we have also calculated a number of other
quantities that could be calculated simultaneously. Our calculations of the
string tension were primarily intended to provide a scale in which to express
our glueball masses. However they also provided a scale for the lattice
spacing $a$ at each value of the bare lattice coupling, $g_{L}(a)$, which we
were able to use to obtain values of the dynamical scale
$\Lambda_{\overline{MS}}$ for all our values of $N$. These improved upon
earlier calculations of this kind and for $SU(3)$ provided a value in physical
units of $\Lambda_{\overline{MS}}\simeq 263(4)[9]{\mathrm{MeV}}$ which is
consistent with values obtained using more dedicated methods At the same time
we were able to confirm that keeping fixed the running ’t Hooft coupling
$g^{2}(a)N$, with $a$ being kept fixed in units of the string tension, is the
way to approach the $N=\infty$ limit along lines of constant physics.

In addition to the fundamental string tension we calculate the tension of
$k=2$ flux tubes, in order to analyse the way that $\sigma_{k=2}/\sigma$
approaches $N=\infty$. We find that a leading $O(1/N)$ correction works better
than the $O(1/N^{2})$ expected from standard large-$N$ counting, but that the
latter cannot be completely excluded. We speculate that the $O(1/N)$ behaviour
is sub-asymptotic, with the $O(1/N^{2})$ correction settling in for flux tubes
of length $l\leq l_{c}$ once $N$ is large enough that the $k=2$ flux tube
becomes a pair of weakly interacting fundamental flux tubes for $l\leq l_{c}$
with $l_{c}\uparrow$ as $N\uparrow$. That is to say, the $l,N\to\infty$ limit
is not uniform.

Since for our glueball calculations we need to monitor the onset of the
freezing of the topology of our lattice fields, we have performed extensive
calculations of the topological charge along with the glueball calculations.
Using these calculations we obtain values for the continuum limit of the
topological susceptibility for the $SU(N\leq 6)$ gauge theories. The freezing
of topology means that we have no values for $N>6$ or for our smaller $SU(6)$
lattice spacings. Nonetheless this does not prevent us achieving a usefully
precise value of the $N=\infty$ topological suseptibility,
$\chi_{I}^{\frac{1}{4}}/\surd\sigma=0.3681(28)$. We have also calculated the
renormalisation factor $Z_{Q}(\beta)$ that relates the value of our particular
(but standard) lattice topological charge, as obtained on the Monte carlo
generated lattice gauge fields, to the true integer valued topological charge
of those fields. This is useful if one wishes to include a topological
$\theta$-term in the action, and so we also include functions that interpolate
between our values of $\beta$. We can do so for all our $SU(N)$ groups because
these calculations can be equally well determined using ensembles of fields
where the topological charge has been inserted through the initial fields of
the Markovian chains.

The present study could be improved in several ways. A definitive study of
ditorelon states and the lightest multiglueball states for all $R^{PC}$
sectors would be useful. The heaviest states need a finer resolution in the
correlation functions for the mass identification to become completely
unambiguous: this could be achieved by using an anisotropic lattice such that
the timelike lattice spacing is much smaller than the spacelike one, a
technique that has occasionally been put to good purpose in the past [50, 51,
7, 10]. Perhaps most important would be the incorporation of more effective
techniques for determining the continuum spins of the glueball states, as for
example in [54] for 2 space dimensions and in [9] for our case here of 3
spatial dimensions.

## Acknowledgements

AA has been financially supported by the European Union’s Horizon 2020
research and innovation programme “Tips in SCQFT” under the Marie Skłodowska-
Curie grant agreement No. 791122. MT acknowledges support by Oxford
Theoretical Physics and All Souls College. The numerical computations were
carried out on the computing cluster in Oxford Theoretical Physics.

##  Appendix A Lattice running couplings

For pure gauge theories the perturbative $\beta$ function for the running
coupling in a coupling scheme $s$ is given by

$\beta(g_{s})=-a\frac{\partial g_{s}}{\partial
a}=-b_{0}g^{3}_{s}-b_{1}g^{5}_{s}-b^{s}_{2}g^{7}_{s}+...,$ (47)

where $a$ is the length scale on which the coupling is calculated and on which
it depends. The first two coefficients $b_{0}$ and $b_{1}$ are scheme
independent while the coefficients $b^{s}_{j\geq 2}$ depend on the scheme.
Integrating between scales $a_{0}$ and $a$, we obtain

$\frac{a}{a_{0}}=\exp{\left(-\int^{g(a)}_{g(a_{0})}\frac{dg}{\beta(g)}\right)}.$
(48)

(The label $s$ on $g$ is to be understood.) The integrand is singular as $g\to
0$ and for any calculations it is convenient to separate out the singular
pieces. Since the issue arise for $g\to 0$ we can expand $1/\beta(g)$ in
powers of $g$ and we then readily see that we can separate out the singular
terms as follows,

$\displaystyle\frac{a}{a_{0}}$ $\displaystyle=$ $\displaystyle
e^{+\int^{g(a)}_{g(a_{0})}dg\left(-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}e^{-\int^{g(a)}_{g(a_{0})}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}$
(49) $\displaystyle=$
$\displaystyle\left(\frac{g^{2}(a)}{g^{2}(a_{0})}\right)^{-\frac{b_{1}}{2b^{2}_{0}}}e^{-\frac{1}{2b_{0}}\left(\frac{1}{g^{2}(a)}-\frac{1}{g^{2}(a_{0})}\right)}e^{-\int^{g(a)}_{g(a_{0})}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}.$

In the second line we have integrated the ‘singular’ terms, and the remaining
integral will now be finite as $g(a_{0})\to 0$. So we can break up the
integral as

$e^{-\int^{g(a)}_{g(a_{0})}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}=e^{+\int^{g(a_{0})}_{0}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}e^{-\int^{g(a)}_{0}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}$
(50)

where each integral will be well-defined since the singularities at $g=0$ have
been removed. Separating the terms in $a$ and $a_{0}$ in eqns(49,50) we can
write

$\frac{a}{a_{0}}=\frac{F(g(a))}{F(g(a_{0}))},$ (51)

where we define

$F(g)\equiv\left(b_{0}g^{2}\right)^{-\frac{b_{1}}{2b^{2}_{0}}}e^{-\frac{1}{2b_{0}g^{2}}}e^{-\int^{g}_{0}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}.$
(52)

Note that the factor of $b_{0}$ that we have inserted in the first term on the
right of eqn(52) will cancel in eqn(51) so we are free to insert it if we
wish. We see from eqn(51) that $a/F(g^{2}(a))$ is independent of the scale $a$
on which the coupling is defined and is a constant. So we can now define a
dynamical energy scale $\Lambda$ by

$\displaystyle\Lambda\equiv\frac{a_{0}}{F(g(a_{0}))}\Longrightarrow a$
$\displaystyle=$ $\displaystyle\frac{1}{\Lambda}F(g(a))$ (53) $\displaystyle=$
$\displaystyle\frac{1}{\Lambda}\left(b_{0}g^{2}\right)^{-\frac{b_{1}}{2b^{2}_{0}}}e^{-\frac{1}{2b_{0}g^{2}}}e^{-\int^{g(a)}_{0}dg\left(\frac{1}{\beta(g)}-\frac{b_{1}}{b^{2}_{0}g}+\frac{1}{b_{0}g^{3}}\right)}.$

The scale $\Lambda$ defined here coincides with the conventional $\Lambda$
parameter that appears in the standard 2-loop expression for the running
coupling. (It is to ensure this equality that we inserted the factor of
$b_{0}$ above.) This scale will clearly depend on the coupling scheme and
within a given scheme the value we obtain for it will depend on our
approximation to $\beta(g)$.

For our lattice action only the first 3 coefficients in the $\beta$-function
are known. In that case, collecting terms, it is convenient to rewrite eqn(53)
as

$a\stackrel{{\scriptstyle
3loop}}{{=}}\frac{1}{\Lambda}\left(b_{0}g^{2}(a)\right)^{-\frac{b_{1}}{2b^{2}_{0}}}e^{-\frac{1}{2b_{0}g^{2}(a)}}e^{-\frac{1}{2}\int^{g^{2}(a)}_{0}dg^{2}\left(\frac{b_{0}b_{2}-b^{2}_{1}-b_{1}b_{2}g^{2}}{b^{3}_{0}+b^{2}_{0}b_{1}g^{2}+b^{2}_{0}b_{2}g^{4}}\right)}\equiv\frac{1}{\Lambda}F_{3l}(g(a)),$
(54)

where we denote by $F_{3l}(g)$ the 3-loop approximation to $F(g)$. The
integrand is a smooth function of $g^{2}$ and so the integral can be
calculated accurately for any given $g(a)$ using any elementary numerical
integration method. If we retain only the first 2 coefficients of $\beta(g)$
then we can do the integral analytically to obtain

$a\stackrel{{\scriptstyle
2loop}}{{=}}\frac{1}{\Lambda}e^{-\frac{1}{2b_{0}g(a)^{2}}}\left(\frac{b_{1}}{b^{2}_{0}}+\frac{1}{b_{0}g^{2}(a)}\right)^{\frac{b_{1}}{2b^{2}_{0}}}\equiv\frac{1}{\Lambda}F_{2l}(g(a))$
(55)

as the exact 2-loop result, where we denote by $F_{2l}(g)$ the 2-loop
approximation to $F(g)$.

The above perturbative expressions for $a\Lambda$ can be turned into
expressions for $a\mu$ where $\mu$ is some physical mass or energy:

$a\mu=\left.a\Lambda\frac{\mu}{\Lambda}\right|_{a}=\left.\frac{\mu}{\Lambda}\right|_{a=0}(1+c_{\mu}a^{2}\mu^{2}+O(a^{4}))F(g(a)),$
(56)

where $F(g(a))$ is defined in eqn(52) and $c_{\mu}$ is an unknown constant.
Here we use the standard tree-level expansion for a dimensionless ratio of
physical energy scales, which here is $\mu/\Lambda$. This expression marries
perturbative (logarithmic) and power-like dependences on $a$ in a plausible
way. It is of course arguable: for example the perturbative expansion is at
best asymptotic and this can introduce other power-like corrections. In
practice we shall use this for the string tension, $\mu=\surd\sigma$, and we
will drop $O(a^{4})$ and higher order terms. That is to say we will attempt to
fit the calculated string tensions with

$a\surd\sigma=\left.\frac{\surd\sigma}{\Lambda}\right|_{a=0}(1+c_{\sigma}a^{2}\sigma)F_{3l}(g(a))$
(57)

and we shall be doing so in the mean-field coupling scheme $s=I$. To calculate
$F_{3l}(g(a))$ in that scheme we only need to know the coefficient
$b_{2}^{s=I}$, since $b_{0}$ and $b_{1}$ are scheme independent with values

$b_{0}={\frac{1}{(4\pi)^{2}}}{\frac{11}{3}}N,\quad
b_{1}={\frac{1}{(4\pi)^{4}}}{\frac{34}{3}}N^{2}.$ (58)

We can begin with the well-known value of $b_{2}^{s=\overline{MS}}$ [72]

$b^{\overline{MS}}_{2}={\frac{1}{(4\pi)^{6}}}{\frac{2857}{54}}N^{3}.$ (59)

To obtain $b^{s}_{2}$ in the improved lattice coupling scheme $I$ we first
transform to the plaquette action lattice coupling scheme, $s=L$, using [73]

$b^{L}_{2}=2b^{\overline{MS}}_{2}-b_{1}l_{0}+b_{0}l_{1},$ (60)

where

$l_{0}=\frac{1}{8N}-0.16995599N,\qquad
l_{1}=-\frac{3}{128N^{2}}+0.018127763-0.0079101185N^{2}.$ (61)

The lattice coupling, $g^{2}_{L}$, satisfies the $\beta$-function in eqn(47)
with $s=L$, and the mean-field coupling will satisfy a $\beta$-function with
the same $b_{0}$ and $b_{1}$ coefficients but with a different coefficient,
$b^{I}_{2}$, of the $g^{7}$ power. To determine $b^{I}_{2}$ one can use the
expression for $\langle\mathrm{Tr}U_{p}\rangle$ as a power series in
$g^{2}_{L}$ to write $g^{2}_{I}$ as

$g^{2}_{I}=g^{2}_{L}\langle\frac{1}{N}\mathrm{Tr}U_{p}\rangle=g^{2}_{L}\left(1-w_{1}g^{2}_{L}-w_{2}g^{4}_{L}-...\right),$
(62)

where [74]

$w_{1}=\frac{(N^{2}-1)}{8N},\qquad
w_{2}=\left(N^{2}-1\right)\left(0.0051069297-\frac{1}{128N^{2}}\right).$ (63)

We now insert the expression for $g^{2}_{I}$ in eqn(62) into the
$\beta$-function for $g^{2}_{L}$ in eqn(47) giving us, after some elementary
manipulation,

$b^{I}_{2}=b^{L}_{2}+w_{2}b_{0}-w_{1}b_{1}.$ (64)

So: using eqns(59,61) in eqn(60) we obtain the explicit expression for
$b^{L}_{2}$ and then inserting that together with the functions in eqn(63) and
eqn(58) into eqn(64) we obtain the explicit expression for $b^{I}_{2}$ for any
$N$. We now have the explicit expressions for $b_{0}$, $b_{1}$ and $b^{I}_{2}$
which allow us to calculate the value of $F_{3l}(g_{I}(a))$ in eqn(57) for any
value of $N$ and $g_{I}(a)$.

##  Appendix B Scattering states

We will restrict ourselves to states of two glueballs. We probe such states
with product operators $\phi_{a}(t)\phi_{b}(t)$ where $\phi_{a}$ and
$\phi_{b}$ are typical single loop zero momentum operators that are expected
to project primarily onto single glueballs with chosen quantum numbers. Both
the individual and product operators will in general need vacuum subtraction
for this to be the case. As usual we will have some basis of single loop
operators so the generic correlator will be of the form

$C_{4}(t)=\langle\phi^{\prime}_{a}(t)\phi^{\prime}_{b}(t)\phi^{\prime}_{c}(0)\phi^{\prime}_{d}(0)\rangle-\langle\phi^{\prime}_{a}\phi^{\prime}_{b}\rangle\langle\phi^{\prime}_{c}\phi^{\prime}_{d}\rangle,$
(65)

where $\phi^{\prime}_{a}(t)=\phi_{a}(t)-\langle\phi_{a}\rangle$ etc. This
equation subtracts any vacuum contribution to the individual operators as well
as to their products. We have taken the operators to be real. If, for example,
we were to include operators $\phi_{i}(t),\phi_{j}(t)$ with opposite non-zero
momenta then these would be complex. In that case we should change
$\phi_{a}(t),\phi_{b}(t)\to\phi^{\dagger}_{a}(t),\phi^{\dagger}_{b}(t)$ in
eqn(65) and below.

Our glueball calculations require high statistics so we calculate our
correlators during the generation of the sequence of lattice fields. At this
stage we can only calculate the correlators of the fields $\phi$ without any
vacuum subtraction – we will only be able to calculate the vacuum expectation
values at the end of the computer simulation. A short calculation tells us
that the correlator of the $\phi^{\prime}_{i}$ fields in eqn(65) can be
written in terms of the correlators of the unsubtracted $\phi_{i}$ fields as
follows:

$C_{4}(t)=\langle\phi_{a}(t)\phi_{b}(t)\phi_{c}(0)\phi_{d}(0)\rangle\\\
-\langle\phi_{a}\rangle\langle\phi_{b}(t)\phi_{c}(0)\phi_{d}(0)\rangle-\langle\phi_{b}\rangle\langle\phi_{a}(t)\phi_{c}(0)\phi_{d}(0)\rangle\\\
-\langle\phi_{c}\rangle\langle\phi_{a}(t)\phi_{b}(t)\phi_{d}(0)\rangle-\langle\phi_{d}\rangle\langle\phi_{a}(t)\phi_{b}(t)\phi_{c}(0)\rangle\\\
+\langle\phi_{a}\rangle\langle\phi_{c}\rangle\langle\phi_{b}(t)\phi_{d}(0)\rangle+\langle\phi_{b}\rangle\langle\phi_{c}\rangle\langle\phi_{a}(t)\phi_{d}(0)\rangle\\\
+\langle\phi_{a}\rangle\langle\phi_{d}\rangle\langle\phi_{b}(t)\phi_{c}(0)\rangle+\langle\phi_{b}\rangle\langle\phi_{d}\rangle\langle\phi_{a}(t)\phi_{c}(0)\rangle\\\
-\langle\phi_{a}\phi_{b}\rangle\langle\phi_{c}\phi_{d}\rangle+2\langle\phi_{a}\phi_{b}\rangle\langle\phi_{c}\rangle\langle\phi_{d}\rangle+2\langle\phi_{c}\phi_{d}\rangle\langle\phi_{a}\rangle\langle\phi_{b}\rangle-4\langle\phi_{a}\rangle\langle\phi_{b}\rangle\langle\phi_{c}\rangle\langle\phi_{d}\rangle.$
(66)

Of course when some of the operators have non-vacuum quantum numbers, then the
corresponding vacuum expectation values will vanish and the above expression
will simplify in obvious ways.

In addition to the above we may also be interested in the overlap of single
loop operators, which mainly project onto single glueballs, with the above
product operators, which one expects to mainly project onto two glueballs.
That is to say correlators such as

$C_{3}(t)=\langle\phi^{\prime}_{a}(t)\phi^{\prime}_{b}(t)\phi^{\prime}_{c}(0)\rangle-\langle\phi^{\prime}_{a}\phi^{\prime}_{b}\rangle\langle\phi^{\prime}_{c}\rangle=\langle\phi^{\prime}_{a}(t)\phi^{\prime}_{b}(t)\phi^{\prime}_{c}(0)\rangle$
(67)

since $\langle\phi^{\prime}_{c}\rangle=0$ by definition. In terms of the
unsubtracted operators we find

$C_{3}(t)=\langle\phi_{a}(t)\phi_{b}(t)\phi_{c}(0)\rangle-\langle\phi_{a}\rangle\langle\phi_{b}(t)\phi_{c}(0)\rangle-\langle\phi_{b}\rangle\langle\phi_{a}(t)\phi_{c}(0)\rangle\\\
-\langle\phi_{c}\rangle\langle\phi_{a}\phi_{b}\rangle+2\langle\phi_{a}\rangle\langle\phi_{b}\rangle\langle\phi_{c}\rangle.$
(68)

In this paper we can only calculate light masses with any reliability so we
restrict ourselves to double loop operators where one loop is in the
$A_{1}^{++}$ representation and therefore has some projection onto the
lightest glueball. The second operator will then be in the representation
$R^{PC}$ in which we happen to be interested. If $R^{PC}\neq A_{1}^{++}$ then
eqn(66) simplifies drastically

$C_{4}(t)=\langle\phi_{a}(t)\phi_{b}(t)\phi_{c}(0)\phi_{d}(0)\rangle\\\
-\langle\phi_{a}\rangle\langle\phi_{b}(t)\phi_{c}(0)\phi_{d}(0)\rangle-\langle\phi_{c}\rangle\langle\phi_{a}(t)\phi_{b}(t)\phi_{d}(0)\rangle+\langle\phi_{a}\rangle\langle\phi_{c}\rangle\langle\phi_{b}(t)\phi_{d}(0)\rangle-\langle\phi_{a}\phi_{b}\rangle\langle\phi_{c}\phi_{d}\rangle$
(69)

since with $\phi_{a},\phi_{c}$ being $A_{1}^{++}$ and $\phi_{b},\phi_{d}$ not
being $A_{1}^{++}$ means that not only
$\langle\phi_{b}\rangle=\langle\phi_{d}\rangle=0$ but also that products like
$\langle\phi_{a}\phi_{b}\rangle$ are zero. Similarly if $\phi_{a}$ in eqn(67)
is $A_{1}^{++}$ and $\phi_{b}$ is in some $R^{PC}\neq A_{1}^{++}$ then
$\phi_{c}$ must be in the same $R^{PC}\neq A_{1}^{++}$ for $C_{3}(t)$ not to
be zero. In that case we will have

$C_{3}(t)=\langle\phi_{a}(t)\phi_{b}(t)\phi_{c}(0)\rangle-\langle\phi_{a}\rangle\langle\phi_{b}(t)\phi_{c}(0)\rangle.$
(70)

If however $\phi_{b}$ and $\phi_{c}$ are in $A_{1}^{++}$ then we need the full
expression.

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SU(2) ; $l\surd\sigma\sim 4$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{2}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
2.2986 | $12^{3}16$ | 0.6018259(46) | 0.36778(69) | 1.224(16)
2.3714 | $14^{3}16$ | 0.6226998(36) | 0.29023(50) | 1.025(12)
2.427 | $20^{3}16$ | 0.6364293(15) | 0.24013(41) | 0.8469(76)
2.509 | $22^{3}20$ | 0.6537214(10) | 0.18011(22) | 0.6563(56)
2.60 | $30^{4}$ | 0.6700089(5) | 0.13283(30) | 0.5001(41)
2.70 | $40^{4}$ | 0.6855713(3) | 0.09737(23) | 0.3652(35)
Table 1: Parameters of the main $SU(2)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(3) ; $l\surd\sigma\sim 4$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{3}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
5.6924 | $10^{3}16$ | 0.5475112(71) | 0.3999(58) | 0.987(9)
5.80 | $12^{3}16$ | 0.5676412(36) | 0.31666(66) | 0.908(12)
5.8941 | $14^{3}16$ | 0.5810697(18) | 0.26118(37) | 0.7991(92)
5.99 | $18^{4}$ | 0.5925636(11) | 0.21982(77) | 0.7045(65)
6.0625 | $20^{4}$ | 0.6003336(10) | 0.19472(54) | 0.6365(43)
6.235 | $26^{4}$ | 0.6167723(6) | 0.15003(30) | 0.4969(29)
6.3380 | $30^{4}$ | 0.6255952(4) | 0.12928(27) | 0.4276(37)
6.50 | $38^{4}$ | 0.6383531(3) | 0.10383(24) | 0.3474(22)
Table 2: Parameters of the main $SU(3)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(4) ; $l\surd\sigma\sim 4$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{4}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
10.70 | $12^{3}16$ | 0.5540665(24) | 0.3021(5) | 0.8406(48)
10.85 | $14^{3}20$ | 0.5664268(15) | 0.25426(38) | 0.7611(54)
11.02 | $18^{3}20$ | 0.5782610(11) | 0.21434(28) | 0.6605(33)
11.20 | $22^{4}$ | 0.5893298(6) | 0.18149(49) | 0.5709(34)
11.40 | $26^{4}$ | 0.6004374(4) | 0.15305(34) | 0.4864(30)
11.60 | $30^{4}$ | 0.6106057(3) | 0.13065(21) | 0.4132(44)
Table 3: Parameters of the main $SU(4)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(5) ; $l\surd\sigma\sim 3.1$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{5}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
16.98 | $10^{3}16$ | 0.5454873(28) | 0.3033(8) | 0.8241(68)
17.22 | $12^{3}16$ | 0.5587002(18) | 0.2546(6) | 0.7517(51)
17.43 | $14^{3}20$ | 0.5685281(10) | 0.22217(37) | 0.6751(44)
17.63 | $16^{3}20$ | 0.5769707(9) | 0.19636(35) | 0.5961(79)
18.04 | $20^{3}24$ | 0.5924012(6) | 0.15622(38) | 0.4783(44)
18.375 | $24^{3}30$ | 0.6036547(4) | 0.13106(30) | 0.4078(38)
Table 4: Parameters of the main $SU(5)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(6) ; $l\surd\sigma\sim 3.1$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{6}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
24.67 | $10^{3}16$ | 0.5409011(28) | 0.30658(34) | 0.8240(41)
25.05 | $12^{3}16$ | 0.5557062(13) | 0.25177(23) | 0.7395(50)
25.32 | $14^{3}20$ | 0.5646185(10) | 0.22208(35) | 0.6673(32)
25.55 | $16^{3}20$ | 0.5715585(9) | 0.20153(34) | 0.6112(41)
26.22 | $20^{3}24$ | 0.5894540(4) | 0.15480(36) | 0.4751(53)
26.71 | $24^{3}30$ | 0.6009861(3) | 0.12867(27) | 0.3886(37)
Table 5: Parameters of the main $SU(6)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(8) ; $l\surd\sigma\sim 2.6$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{8}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
44.10 | $8^{3}16$ | 0.5318034(31) | 0.32589(62) | 0.8246(66)
44.85 | $10^{3}16$ | 0.5497960(15) | 0.25791(40) | 0.7461(53)
45.50 | $12^{3}20$ | 0.5622253(9) | 0.21851(45) | 0.6409(38)
46.10 | $14^{3}20$ | 0.5723242(8) | 0.18932(38) | 0.5617(43)
46.70 | $16^{3}24$ | 0.5815072(5) | 0.16557(38) | 0.4909(43)
47.75 | $20^{3}30$ | 0.5959878(3) | 0.13253(26) | 0.4075(28)
Table 6: Parameters of the main $SU(8)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(10) ; $l\surd\sigma\sim 2.6$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{10}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
69.20 | $8^{3}16$ | 0.5292925(29) | 0.33024(35) | 0.8282(60)
70.38 | $10^{3}16$ | 0.5478565(13) | 0.25987(30) | 0.7435(42)
71.38 | $12^{3}20$ | 0.5602903(9) | 0.21988(32) | 0.6451(44)
72.40 | $14^{3}20$ | 0.5713707(6) | 0.18845(20) | 0.5549(59)
73.35 | $16^{3}24$ | 0.5807004(4) | 0.16399(19) | 0.4952(44)
Table 7: Parameters of the main $SU(10)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. SU(12) ; $l\surd\sigma\sim 2.6$
---
$\beta$ | $l^{3}l_{t}$ | $\tfrac{1}{12}\text{ReTr}\langle U_{p}\rangle$ | $a\surd\sigma$ | $am_{G}$
99.86 | $8^{3}16$ | 0.5275951(27) | 0.33341(40) | 0.8243(52)
101.55 | $10^{3}16$ | 0.5464461(12) | 0.26162(32) | 0.7384(51)
103.03 | $12^{3}20$ | 0.55936304(61) | 0.21915(25) | 0.6432(32)
104.55 | $14^{3}20$ | 0.57087665(49) | 0.18663(38) | 0.5521(42)
105.95 | $16^{3}24$ | 0.58043063(32) | 0.16197(27) | 0.4920(37)
Table 8: Parameters of the main $SU(12)$ calculations: the inverse coupling,
$\beta$, the lattice size, the average plaquette, the string tension,
$\sigma$, and the mass gap, $m_{G}$. $aE_{eff}(t=2a)$ : $SU(8)$ at
$\beta=45.50$ on $14^{3}20$
---
$R^{PC}$ | all $Q$ | $Q=0$ | $R^{PC}$ | all $Q$ | $Q=0$
$A_{1}^{++}$ | 0.6512(21) | 0.6497(25) | $A_{1}^{-+}$ | 1.119(8) | 1.098(9)
| 1.199(9) | 1.209(9) |  | 1.581(23) | 1.587(21)
| 1.243(12) | 1.236(11) |  | 1.908(43) | 1.937(51)
| 1.565(21) | 1.551(23) |  |  |
$A_{2}^{++}$ | 1.668(27) | 1.642(25) | $A_{2}^{-+}$ | 2.21(8) | 2.51(11)
| 1.983(48) | 2.037(48) |  | 2.30(9) | 2.23(11)
$E^{++}$ | 1.020(4) | 1.020(5) | $E^{-+}$ | 1.354(9) | 1.356(11)
| 1.230(8) | 1.233(8) |  | 1.757(24) | 1.765(25)
| 1.435(10) | 1.424(9) |  | 2.060(51) | 2.157(50)
$T_{1}^{++}$ | 1.670(14) | 1.673(18) | $T_{1}^{-+}$ | 1.841(25) | 1.861(24)
| 1.728(20) | 1.708(19) |  | 1.976(27) | 1.950(26)
| 2.011(27) | 2.041(35) |  | 1.925(28) | 1.928(28)
| 2.053(35) | 2.037(31) |  |  |
$T_{2}^{++}$ | 1.032(4) | 1.033(3) | $T_{2}^{-+}$ | 1.360(8) | 1.356(9)
| 1.460(9) | 1.447(8) |  | 1.727(15) | 1.714(17)
| 1.670(15) | 1.648(18) |  | 1.881(20) | 1.900(24)
| 1.692(14) | 1.691(17) |  |  |
$A_{1}^{+-}$ | 2.10(7) | 2.10(8) | $A_{1}^{--}$ | 2.10(7) | 2.10(8)
| 2.33(9) | 2.41(12) |  | 2.41(12) | 2.40(13)
$A_{2}^{+-}$ | 1.557(18) | 1.577(21) | $A_{2}^{--}$ | 2.00(5) | 2.06(7)
| 1.852(29) | 1.754(26) |  | 2.15(8) | 2.25(9)
| 2.13(6) | 2.27(10) |  |  |
$E^{+-}$ | 1.980(35) | 1.981(33) | $E^{--}$ | 1.690(18) | 1.686(21)
| 2.155(48) | 2.119(41) |  | 2.072(39) | 1.998(42)
| 2.198(54) | 2.087(50) |  | 2.198(57) | 2.204(66)
$T_{1}^{+-}$ | 1.266(5) | 1.271(6) | $T_{1}^{--}$ | 1.738(18) | 1.734(18)
| 1.542(9) | 1.557(10) |  | 1.973(29) | 1.920(26)
| 1.656(13) | 1.648(11) |  | 2.088(32) | 2.001(31)
| 1.859(18) | 1.860(19) |  |  |
$T_{2}^{+-}$ | 1.571(11) | 1.543(11) | $T_{2}^{--}$ | 1.721(15) | 1.715(21)
| 1.880(19) | 1.881(20) |  | 1.888(23) | 1.889(28)
| 1.900(20) | 1.915(27) |  | 2.043(27) | 2.006(26)
| 1.996(34) | 1.994(30) |  |  |
$l_{k=1}$ | 0.5963(11) | 0.5932(14) | $l_{k=2}$ | 1.1204(33) | 1.1235(28)
| 1.2954(33) | 1.2918(42) |  | 1.3876(57) | 1.3810(64)
Table 9: Comparison of glueball and flux tube energies obtained on fields with
topological charge $Q=0$ against fields with a ‘normal’ distribution of $Q$.
In $SU(8)$ on a $14^{3}20$ lattice at $\beta=45.50$. Energies extracted from
best (variationally selected) correlators between $t=a$ and $t=2a$. Glueballs
labelled by representation of cubic rotation symmetry $R$, parity $P$ and
charge conjugation $C$. Flux tubes are fundamental, $l_{k=1}$, and $k=2$,
$l_{k=2}$.

SU(2) $\beta=2.427$ | SU(3) $\beta=6.235$ | SU(6) $\beta=25.35$
---|---|---
$l$ | $a\surd\sigma$ | $l$ | $a\surd\sigma$ | $l$ | $a\surd\sigma$
12 | 0.2380(3) | 18 | 0.1491(5) | 12 | 0.2196(5)
14 | 0.2387(4) | 26 | 0.1499(4) | 14 | 0.2200(7)
16 | 0.2390(8) | 34 | 0.1506(6) | 18 | 0.2181(17)
20 | 0.2399(10) |  |  |  |
24 | 0.2396(16) |  |  |  |
SU(4) $\beta=11.02$ | SU(4) $\beta=11.60$ | SU(8) $\beta=45.50$
$l$ | $a\surd\sigma$ | $l$ | $a\surd\sigma$ | $l$ | $a\surd\sigma$
18 | 0.2143(3) | 24 | 0.1301(5) | 12 | 0.2187(4)
22 | 0.2142(6) | 30 | 0.1307(3) | 14 | 0.2189(4)
SU(10) $\beta=71.38$ | SU(12) $\beta=103.03$ |
$l$ | $a\surd\sigma$ | $l$ | $a\surd\sigma$ |  |
12 | 0.2199(4) | 12 | 0.2198(2) |  |
14 | 0.2195(5) | 14 | 0.2202(7) |  |
Table 10: String tensions obtained using eqn(8), for (fundamental) flux tubes of length $l$ for various groups. A test of finite volume corrections. SU(4) $\beta=11.02$ | SU(4) $\beta=11.60$ | SU(6) $\beta=25.35$
---|---|---
$l$ | $a\surd\sigma_{k=2}$ | $l$ | $a\surd\sigma_{k=2}$ | $l$ | $a\surd\sigma_{k=2}$
18 | 0.2490(12) | 24 | 0.1523(5) | 12 | 0.2779(9)
22 | 0.2480(24) | 30 | 0.1534(5) | 14 | 0.2846(6)
|  |  |  | 18 | 0.2841(8)
SU(8) $\beta=45.50$ | SU(10) $\beta=71.38$ | SU(12) $\beta=103.03$
$l$ | $a\surd\sigma_{k=2}$ | $l$ | $a\surd\sigma_{k=2}$ | $l$ | $a\surd\sigma_{k=2}$
12 | 0.2833(9) | 12 | 0.2901(10) | 12 | 0.2925(11)
14 | 0.2885(24) | 14 | 0.2946(22) | 14 | 0.2987(23)
Table 11: String tensions obtained using eqn(8) for $k=2$ flux tubes of length
$l$ for various groups. Testing finite volume corrections. SU(2)
---
$\beta$ | lattice | $\tfrac{1}{2}\text{ReTr}\langle U_{p}\rangle$ | $aE_{f}$ | $a\surd\sigma$
2.2986 | $8^{3}16$ | 0.6018323(78) | 0.9310(55) | 0.36590(93)
2.3714 | $10^{3}16$ | 0.6227129(41) | 0.7159(41) | 0.28779(71)
2.427 | $12^{3}16$ | 0.6364295(26) | 0.5831(32) | 0.23750(56)
2.452 | $14^{3}20$ | 0.6420346(28) | 0.5852(51) | 0.21791(83)
2.509 | $16^{3}20$ | 0.6537206(12) | 0.4399(15) | 0.17857(26)
2.60 | $22^{3}30$ | 0.6700085(8) | 0.3370(17) | 0.13279(29)
2.65 | $26^{3}34$ | 0.6780431(5) | 0.2914(15) | 0.11342(26)
2.70 | $28^{3}40$ | 0.6855710(4) | 0.2228(13) | 0.09698(24)
2.75 | $34^{3}46$ | 0.6926656(2) | 0.2048(16) | 0.08375(21)
2.80 | $40^{3}54$ | 0.6993804(2) | 0.1817(16) | 0.07242(28)
Table 12: Energies of periodic flux tubes of length $l$ in $SU(2)$ and derived
string tensions at the couplings $\beta=4/g^{2}$ on the lattices $l^{3}l_{t}$.
SU(3)
---
$\beta$ | lattice | $\tfrac{1}{3}\text{ReTr}\langle U_{p}\rangle$ | $aE_{f}$ | $a\surd\sigma$
5.6924 | $8^{3}16$ | 0.547503(7) | 1.1588(37) | 0.4010(23)
5.80 | $10^{3}16$ | 0.567642(5) | 0.8862(26) | 0.31603(84)
5.8941 | $12^{3}16$ | 0.581069(4) | 0.7298(47) | 0.2613(14)
5.99 | $14^{3}20$ | 0.5925655(13) | 0.5984(26) | 0.21959(76)
6.0625 | $14^{3}20$ | 0.6003331(21) | 0.4517(25) | 0.19509(65)
6.235 | $18^{3}26$ | 0.6167715(13) | 0.3369(26) | 0.14899(59)
6.338 | $22^{3}30$ | 0.6255948(8) | 0.3182(19) | 0.12948(44)
6.50 | $26^{3}38$ | 0.6383532(5) | 0.2334(17) | 0.10319(41)
6.60 | $32^{3}40$ | 0.64566194(21) | 0.2255(15) | 0.09024(26)
6.70 | $36^{3}44$ | 0.65260033(18) | 0.1933(16) | 0.07898(28)
Table 13: Energies of periodic flux tubes of length $l$ in $SU(3)$ and derived
string tensions at the couplings $\beta=6/g^{2}$ on the lattices $l^{3}l_{t}$.
continuum $k=2$ string tensions
---
group |  $\sigma_{k}/\sigma_{f}$:NG |  $\sigma_{k}/\sigma_{f}$:$l\to\infty$ | Casimir scaling
$SU(4)$ | 1.381(14) | 1.381(14) | 1.333
$SU(5)$ | 1.551(11) | 1.551(11) | 1.500
$SU(6)$ | 1.654(13) | 1.654(13) | 1.600
$SU(8)$ | 1.731(11) | 1.794(28) | 1.714
$SU(10)$ | 1.733(15) | 1.796(29) | 1.778
$SU(12)$ | 1.792(16) | 1.857(29) | 1.818
Table 14: Ratio of the $k=2$ string tension to the fundamental for various $SU(N)$ gauge theories. Values labelled NG are obtained using eqn(8). Values labelled $l\to\infty$ denote our best estimates in that limit. The values corresponding to ‘Casimir scaling’ are shown for comparison. $N$ | $a\sqrt{\sigma}\in$ | $\sqrt{\sigma}/\Lambda^{3loop}_{I}$ | $\chi^{2}/n_{df}$ | $\sqrt{\sigma}/\Lambda^{2loop}_{I}$ | $\Lambda^{3loop}_{\overline{MS}}/\sqrt{\sigma}$
---|---|---|---|---|---
2 | [0.133,0.072] | 4.535(16) | 0.42 | 4.914(16) | 0.5806(21)[570]
3 | [0.195,0.079] | 4.855(11) | 1.40 | 5.210(20) | 0.5424(13)[185]
4 | [0.254,0.131] | 5.043(10) | 0.48 | 5.532(11) | 0.5222(11)[230]
5 | [0.255,0.131] | 5.090(14) | 0.31 | 5.622(15) | 0.5174(15)[245]
6 | [0.252,0.129] | 5.105(11) | 1.62 | 5.664(48)∗ | 0.5158(11)[250]
8 | [0.258,0.133] | 5.148(17) | 0.13 | 5.727(20) | 0.5115(17)[250]
10 | [0.260,0.164] | 5.221(40)∗ | 3.30 | 5.845(44)∗ | 0.5044(20)[270]
12 | [0.262,0.162] | 5.193(13) | 0.45 | 5.823(15) | 0.5071(13)[270]
Table 15: Results for $\Lambda_{I}$ in units of the string tension using the exact 3-loop $\beta$-function, with the 2-loop result for comparison. Poor fits denoted by $\ast$. Resulting $\Lambda_{\overline{MS}}$ is shown with statistical errors and an estimate of the (correlated) systematic error in square brackets. $N$ | $c_{0}$ | $c_{\sigma}$ | $a\sqrt{\sigma}\in$ | $\beta\in$ | $\chi^{2}/n_{df}$ | $n_{df}$
---|---|---|---|---|---|---
2 | 4.510(15) | 4.98(25) | [0.133,0.072] | [2.60,2.80] | 0.40 | 3
3 | 4.827(12) | 2.48(11) | [0.195,0.079] | [5.99,6.70] | 1.29 | 4
4 | 5.017(8) | 1.622(40) | [0.302,0.131] | [10.70,11.60] | 0.98 | 4
5 | 5.068(11) | 1.435(56) | [0.303,0.131] | [16.98,18.375] | 0.38 | 4
6 | 5.064(12) | 1.541(68) | [0.252,0.129] | [25.05,26.71] | 1.45 | 3
8 | 5.143(9) | 1.265(34) | [0.326,0.133] | [44.10,47.75] | 1.75 | 4
10 | 5.176(10) | 1.285(35) | [0.260,0.164] | [70.38,73.35] | 3.06 | 2
12 | 5.148(12) | 1.387(43) | [0.262,0.162] | [101.55,105.95] | 0.41 | 2
Table 16: Fitted values of the coefficients $c_{0}$ and $c_{\sigma}$ that
determine the interpolation function in eqn(25) for each of our $SU(N)$
lattice gauge theories. In each case the range in $a\surd\sigma$ and of
$\beta$ of the fit is shown as is the $\chi^{2}$ per degree of freedom and the
number of degrees of freedom. $aE_{eff}(t=2a)$ : $SU(2)$ at $\beta=2.427$
---
$R^{P}$ | $12^{3}16$ | $14^{3}16$ | $20^{3}16$
$A_{1}^{+}$ | 0.840(11) | 0.854(8) | 0.847(8)
| 1.225(26) |  |
| 1.418(17) | 1.367(15) | 1.408(19)
|  | 1.593(27) |
| 1.611(36) | 1.721(51) | 1.780(45)
| 1.871(40) | 1.839(46) | 1.872(46)
$A_{2}^{+}$ | 1.680(40) | 1.805(38) | 1.942(72)
$E^{+}$ | 1.220(9) | 1.254(14) | 1.284(13)
| 1.317(17) |  |
|  | 1.549(20) |
| 1.632(30) | 1.672(33) | 1.672(40)
| 1.774(31) | 1.820(39) | 1.832(42)
$T_{1}^{+}$ | 1.925(42) | 1.875(27) | 1.855(37)
| 1.977(48) | 1.972(34) | 1.916(50)
$T_{2}^{+}$ | 1.297(12) | 1.278(14) | 1.289(11)
| 1.739(20) | 1.701(30) | 1.672(21)
| 1.774(34) | 1.849(22) | 1.877(30)
$A_{1}^{-}$ | 1.471(30) | 1.540(30) | 1.526(36)
| 1.71(4) | 1.87(9) | 1.94(9)
$A_{2}^{-}$ | 2.31(16) | 2.21(17) | 2.04(12)
$E^{-}$ | 1.585(31) | 1.680(28) | 1.652(34)
| 2.00(6) | 2.04(7) | 2.02(7)
$T_{1}^{-}$ | 2.18(6) | 2.15(6) | 2.09(7)
$T_{2}^{-}$ | 1.665(25) | 1.632(33) | 1.620(21)
| 2.04(5) | 2.00(4) | 2.01(6)
$l_{k=1}$ | 0.5804(38) | 0.7140(41) | 1.0996(39)
| 1.367(23) | 1.428(32) | 1.677(9)
Table 17: Comparison of glueball effective energies from $t=2a$ and flux tube
energies ($l1$) obtained on $12^{3}16$, $14^{3}16$ and $20^{3}16$ lattices at
$\beta=2.427$ in $SU(2)$. Unmatched states are ditorelons. Glueballs labelled
by representation of cubic rotation symmetry $R$ and parity $P$. Fundamental
flux tube energies , $l_{k=1}$, are ground and first excited states.

$aE_{eff}(t=2a)$ : $SU(12)$ at $\beta=103.03$
---
$R^{PC}$ | $12^{3}20$ | $14^{3}20$ | $R^{PC}$ | $12^{3}20$ | $14^{3}20$
$A_{1}^{++}$ | 0.6434(32) | 0.6499(50) | $A_{1}^{-+}$ | 1.079(8) | 1.097(12)
| 1.190(9) | 1.200(14) |  | 1.567(22) | 1.591(33)
| 1.495(19) | 1.579(25) |  | 1.859(38) | 2.037(91)
| 1.604(26) | 1.585(33) |  |  |
| 1.660(26) | 1.687(43) |  |  |
$A_{2}^{++}$ | 1.629(30) | 1.639(39) | $A_{2}^{-+}$ | 2.14(10) | 2.21(11)
| 1.91(5) | 2.11(10) |  | 2.20(9) | 2.30(13)
$E^{++}$ | 1.021(5) | 1.026(7) | $E^{-+}$ | 1.346(10) | 1.355(15)
| 1.429(14) | 1.464(16) |  | 1.723(21) | 1.747(32)
| 1.610(19) | 1.632(28) |  | 2.105(33) | 2.178(74)
$T_{1}^{++}$ | 1.635(17) | 1.692(21) | $T_{1}^{-+}$ | 1.865(21) | 1.885(39)
| 1.712(19) | 1.776(26) |  | 1.945(30) | 1.930(36)
| 2.090(32) | 2.062(47) |  | 2.012(28) | 2.026(49)
$T_{2}^{++}$ | 1.040(4) | 1.031(7) | $T_{2}^{-+}$ | 1.357(8) | 1.366(12)
| 1.465(9) | 1.473(16) |  | 1.730(16) | 1.748(23)
| 1.597(14) | 1.659(21) |  | 1.891(26) | 1.905(44)
| 1.652(14) | 1.676(25) |  | 1.932(29) | 1.975(40)
$A_{1}^{+-}$ | 2.001(52) | 2.06(11) | $A_{1}^{--}$ | 2.18(8) | 2.09(11)
| 2.12(8) | 2.23(13) |  | 2.23(10) | 2.30(12)
$A_{2}^{+-}$ | 1.564(22) | 1.603(30) | $A_{2}^{--}$ | 1.932(52) | 2.026(70)
| 1.823(36) | 1.882(53) |  | 2.209(85) | 2.33(17)
| 1.97(8) | 2.19(10) |  |  |
$E^{+-}$ | 1.903(29) | 2.011(43) | $E^{--}$ | 1.682(17) | 1.677(30)
| 2.140(51) | 2.044(70) |  | 2.053(44) | 2.087(54)
|  |  |  | 2.11(5) | 2.26(9)
$T_{1}^{+-}$ | 1.269(6) | 1.268(8) | $T_{1}^{--}$ | 1.696(18) | 1.693(22)
| 1.543(12) | 1.546(16) |  | 1.921(25) | 1.897(44)
| 1.678(16) | 1.690(18) |  | 1.976(33) | 2.023(45)
| 1.850(21) | 1.880(27) |  | 2.21(6) | 2.18(6)
$T_{2}^{+-}$ | 1.534(13) | 1.541(15) | $T_{2}^{--}$ | 1.719(20) | 1.766(27)
| 1.878(24) | 1.870(27) |  | 1.893(29) | 1.935(31)
| 1.850(22) | 1.896(36) |  | 2.014(32) | 2.035(37)
| 1.988(32) | 1.987(43) |  | 2.19(5) | 2.19(6)
$l_{k=1}$ | 0.4812(13) | 0.5993(40) | $l_{k=2}$ | 0.935(8) | 1.172(18)
Table 18: Comparison of glueball effective energies from $t=2a$ and flux tube
energies ($l1$,$l2$) obtained on $12^{3}20$ and $14^{3}20$ lattices at
$\beta=103.03$ in $SU(12)$. Glueballs labelled by representation of cubic
rotation symmetry $R$, parity $P$ and charge conjugation $C$. Flux tube
energies are fundamental, $l_{k=1}$, and $k=2$, $l_{k=2}$. $aE_{eff}(t=2a)$ :
$SU(12)$ at $\beta=103.03$
---
$R^{PC}$ | $12^{3}20$ | $14^{3}20$
| $bl=1-4$ | $bl=1-5$ | $bl=1-4$ | $bl=1-5$
$A_{1}^{++}$ | 0.6434(32) | 0.6432(32) | 0.6499(50) | 0.6497(50)
|  | 1.172(10) |  |
| 1.190(9) | 1.179(14) | 1.200(14) | 1.200(14)
|  |  |  | 1.424(23)
| 1.462(20) | 1.495(19) | 1.588(23) | 1.433(29)
| 1.604(26) | 1.601(29) | 1.585(33) | 1.571(35)
| 1.660(26) | 1.666(30) | 1.687(43) | 1.659(40)
| 1.753(25) | 1.780(31) | 1.786(54) | 1.780(52)
$E^{++}$ | 1.021(5) | 1.019(5) | 1.026(7) | 1.026(7)
| 1.525(13)[1.01(14)] | 1.118(7) |  |
|  |  |  | 1.273(20)
| 1.429(14) | 1,373(12) | 1.464(16) | 1.446(16)
| 1.610(19) | 1.610(18) | 1.635(29) | 1.632(28)
| 1.657(19) | 1.650(18) | 1.663(24) | 1.666(25)
Table 19: Comparison of $A_{1}^{++}$ and $E^{++}$ glueball effective energies
from $t=2a$ obtained on $12^{3}20$ and $14^{3}20$ lattices at $\beta=103.03$
in $SU(12)$. Using operators up to blocking levels 4 and 5 respectively, as
shown. number of operators: $N\in[4,12]$
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 12 | 7 | 6 | 8
$A_{2}$ | 7 | 5 | 7 | 7
$E$ | 36 | 24 | 24 | 30
$T_{1}$ | 48 | 54 | 66 | 51
$T_{2}$ | 60 | 60 | 60 | 54
number of operators: $N=2,3$
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 27 | 9 | 8 | 11
$A_{2}$ | 14 | 6 | 13 | 11
$E$ | 80 | 30 | 40 | 44
$T_{1}$ | 78 | 84(75) | 132 | 81
$T_{2}$ | 108 | 96(87) | 108 | 84
Table 20: Number of operators in rotational representation , $R$, with parity,
$P$, and charge conjugation, $C$, as used in our various $SU(N)$ calculations,
at each blocking level. Some numbers for $SU(2)$ differ from $SU(3)$ and are
in brackets; also no $C=-$ for $SU(2)$ operator loops
---
$loop$ | $R^{PC}$
{2,3,-2,-3} | $A_{1}^{++},E^{++}$
| $T_{1}^{+-}$
{1,2,2,-1,-2,-2} | $A_{1}^{++},A_{2}^{++},E^{++}$
| $T_{1}^{+-},T_{2}^{+-}$
{1,2,3,-1,-2,-3} | $A_{1}^{++},T_{2}^{++}$
| $A_{2}^{+-},T_{1}^{+-}$
{1,3,2,-3,-1,-2} | $A_{1}^{++},E^{++},T_{2}^{++},T_{1}^{-+},T_{2}^{-+}$
| $T_{1}^{+-},T_{2}^{+-},A_{1}^{--},E^{--},T_{2}^{--}$
{1,2,2,-1,3,-2,-3,-2} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},A_{2}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{1,3,-1,-3,-1,-2,1,2} | $A_{1}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{1,2,3,-1,-3,-3,-2,3} | $A_{1}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{1,3,1,2,-3,-1,-1,-2} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},A_{2}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{1,2,2,2,-1,3,-2,-3,-2,-2} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},A_{2}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{1,2,2,2,-1,-2,3,-2,-3,-2} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},T_{1}^{-+},T_{2}^{-+}$
| $T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{-3,1,3,1,2,-3,-1,3,-1,-2} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},A_{2}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
{-3,1,3,1,2,3,-1,-3,-2,-1} | $A_{1}^{++},A_{2}^{++},E^{++},T_{1}^{++},T_{2}^{++},A_{1}^{-+},A_{2}^{-+},E^{-+},T_{1}^{-+},T_{2}^{-+}$
|
$A_{1}^{+-},A_{2}^{+-},E^{+-},T_{1}^{+-},T_{2}^{+-},A_{1}^{--},A_{2}^{--},E^{--},T_{1}^{--},T_{2}^{--}$
Table 21: The 12 loops used as the basis of our glueball calculations for
$N\geq 4$. These generate contributions to the representations as shown.
$SU(8):aM_{G}$
---
$R^{PC}$ | $\beta=44.10$ | $\beta=44.85$ | $\beta=45.50$ | $\beta=46.10$ | $\beta=46.70$ | $\beta=47.75$
| $8^{3}16$ | $10^{3}16$ | $12^{3}20$ | $14^{3}20$ | $16^{3}24$ | $20^{3}30$
$A^{++}_{1}$ | 0.8246(66) | 0.7461(53) | 0.6409(38) | 0.5617(43) | 0.4909(43) | 0.4075(28)
| 1.615(26) | 1.191(50) | 1.112(37) | 1.081(22) | 0.930(13) | 0.755(10)
| 2.01(6) | 1.764(39) | 1.42(9) | 1.252(41) | 1.146(27) | 0.936(18)
$A^{++}_{2}$ | 2.42(11) | 1.828(46) | 1.615(25) | 1.408(26) | 1.180(27) | 0.969(16)
$E^{++}$ | 1.504(13) | 1.184(8) | 1.0063(51) | 0.8813(48) | 0.7676(67) | 0.6192(69)
| 2.145(41) | 1.689(18) | 1.442(11) | 1.299(34) | 1.070(16) | 0.831(8)
|  | 1.855(28) | 1.579(15) | 1.393(13) | 1.184(22) | 0.954(11)
$T^{++}_{1}$ | 2.50(8) | 1.970(29) | 1.517(27) | 1.397(40) | 1.204(21) | 0.982(7)
|  | 1.932(29) | 1.536(90) | 1,392(43) | 1.238(29) | 0.996(10)
$T^{++}_{2}$ | 1.536(12) | 1.191(22) | 1.033(10) | 0.8795(58) | 0.7725(64) | 0.6195(22)
| 2.092(41) | 1.689(15) | 1.428(39) | 1.188(21) | 1.101(12) | 0.870(16)
| 2.41(8) | 1.875(28) | 1.621(76) | 1.357(34) | 1.162(14) | 0.963(8)
|  | 1.889(30) | 1.52(8) | 1.344(35) | 1.201(18) | 0.975(24)
$A^{-+}_{1}$ | 1.618(33) | 1.256(13) | 1.009(20) | 0.902(17) | 0.8021(95) | 0.6365(81)
| 2.21(10) | 1.75(5) | 1.42(8) | 1.23(5) | 1.074(82) | 0.963(35)
$A^{-+}_{2}$ |  | 2.27(13) | 2.07(6) | 1.857(38) | 1.588(32) | 1.257(33)
$E^{-+}$ | 1.994(56) | 1.573(17) | 1.295(41) | 1.131(21) | 1.013(11) | 0.8082(65)
| 2.72(14) | 2.019(54) | 1.734(20) | 1.498(14) | 1.320(9) | 1.101(15)
$T^{-+}_{1}$ |  | 2.29(15) | 1.844(23) | 1.571(65) | 1.324(32) | 1.108(11)
c |  |  | 1.990(24) | 1.639(12) | 1.447(39) | 1.142(18)
$T^{-+}_{2}$ | 1.959(26) | 1.47(5) | 1.271(24) | 1.146(16) | 0.982(10) | 0.807(10)
| 2.62(11) | 2.043(43) | 1.741(18) | 1.506(11) | 1.298(25) | 1.065(11)
$A^{+-}_{1}$ |  | 2.47(16) | 2.103(73) | 1.831(40) | 1.552(19) | 1.209(32)
$A^{+-}_{2}$ | 2.25(10) | 1.834(37) | 1.529(18) | 1.275(44) | 1.176(26) | 0.928(13)
$E^{+-}$ |  | 2.196(52) | 1.921(33) | 1.659(26) | 1.463(38) | 1.117(18)
$T^{+-}_{1}$ | 1.898(26) | 1.510(12) | 1.281(7) | 1.102(6) | 0.9556(82) | 0.7706(46)
| 2.23(6) | 1.812(27) | 1.539(13) | 1.328(6) | 1.1724(41) | 0.9269(78)
|  | 1.914(21) | 1.648(14) | 1.437(10) | 1.221(16) | 0.980(8)
$T^{+-}_{2}$ | 2.33(6) | 1.781(21) | 1.544(15) | 1.294(30) | 1.133(14) | 0.922(7)
| 2.53(12) | 2.237(46) | 1.811(22) | 1.50(6) | 1.327(32) | 1.093(15)
| 2.53(10) | 2.265(50) | 1.815(20) | 1,618(18) | 1.343(33) | 1.119(13)
$A^{--}_{1}$ |  | 2.62(18) | 2.05(7) | 1.888(35) | 1.633(23) | 1.254(36)
$A^{--}_{2}$ |  | 2.22(10) | 1.94(6) | 1.74(17) | 1.368(67) | 1.114(24)
$E^{--}$ | 2.42(9) | 2.012(40) | 1.663(19) | 1.441(11) | 1.250(31) | 1.009(14)
$T^{--}_{1}$ | 2.55(9) | 1.906(30) | 1.564(69) | 1.414(42) | 1.186(22) | 0.974(29)
$T^{--}_{2}$ | 2.48(7) | 1.981(32) | 1.709(16) | 1.473(10) | 1.2920(56) | 1.037(9)
|  | 2.24(5) | 1.896(22) | 1.601(13) | 1.4108(81) | 1.109(14)
$aE_{f}$ | 0.7067(33) | 0.5506(21) | 0.4778(24) | 0.4204(20) | 0.3674(20) | 0.2943(14)
$aE_{k=2}$ | 1.268(20) | 0.9948(85) | 0.8716(61) | 0.7923(69) | 0.6735(46) | 0.5483(29)
Table 22: $SU(8)$ lattice glueball masses for all $R^{PC}$ representations,
with the fundamental, $aE_{f}$, and $k=2$, $aE_{k=2}$, flux tube energies.

$SU(2)$: $M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+
$A_{1}$ | 3.781(23) | 6.017(61)
| 6.126(38) | 8.00(15)
| 7.54(10) |
$A_{2}$ | 7.77(18) | 9.50(18)⋆
| 8.56(21) |
$E$ | 5.343(30) | 7.037(67)
| 6.967(62) | 8.574(83)
| 7.722(82) | 9.58(16)
$T_{1}$ | 8.14(10) | 9.06(13)
| 8.46(12) | 9.40(13)
| 9.67(9) | 9.83(16)
$T_{2}$ | 5.353(23) | 6.997(65)
| 7.218(52) | 8.468(86)
| 8.23(10) | 9.47(10)
Table 23: $SU(2)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$. Ground states and some excited states. Stars
indicate poor fits (see text). $SU(3)$: $M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.405(21) | 5.276(45) | 9.32(28) | 9.93(49)
| 5.855(41) | 7.29(13) |  | 10.03(47)
| 7.38(11) | 9.18(26) |  |
| 7.515(50) | 9.37(22) |  |
$A_{2}$ | 7.705(85) | 9.80(22) | 7.384(90) | 8.96(15)
| 8.61(20) | 11.17(30) | 8.94(10) | 10.21(20)
|  |  | 8.90(21) |
$E$ | 4.904(20) | 6.211(56) | 8.77(12) | 7.91(10)
| 6.728(47) | 8.23(9) | 9.03(23) | 9.39(18)
| 7.49(9) | 9.47(16) | 10.39(21) | 10.40(22)
| 7.531(60) |  |  |
$T_{1}$ | 7.698(80) | 8.48(12) | 6.065(40) | 8.31(10)
| 7.72(11) | 8.57(13) | 7.21(8) | 9.30(14)⋆⋆
| 9.31(11)⋆ | 8.66(15) | 7.824(56) | 9.72(15)
|  | 9.56(28) | 8.92(10) |
$T_{2}$ | 4.884(19) | 6.393(45) | 7.220(86) | 8.198(80)
| 6.814(31) | 8.15(7) | 8.72(11) | 8.99(11)⋆⋆
| 7.716(70)⋆⋆ | 9.23(12)⋆ | 9.060(80) | 9.69(13)
| 7.677(71) |  |  |
Table 24: $SU(3)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text). $SU(4)$:
$M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.271(27) | 5.020(46) | 9.22(22)⋆ | 10.27(28)
| 5.827(62) | 7.33(11) | 10.43(21) | 9.95(29)
| 7.50(11) | 8.98(23) |  |
| 7.73(8) |  |  |
$A_{2}$ | 7.32(12) | 9.67(19) | 6.87(26) | 8.89(13)
| 8.42(22) | 10.69(45)⋆ | 8.66(16) | 10.99(35)
|  |  | 9.57(19) |
$E$ | 4.721(27) | 6.130(52) | 8.73(20) | 7.80(11)
| 6.702(45) | 7.91(13) | 9.28(19) | 9.55(13)⋆
| 7.271(86) | 9.13(22) | 9.58(20) | 10.06(26)
| 7.586(84) |  |  |
$T_{1}$ | 7.42(12) | 8.47(11) | 5.956(42) | 7.603(84)
| 7.50(9) | 8.59(16) | 7.11(8) | 8.92(12)⋆⋆
| 9.10(18) | 8.62(17)⋆⋆ | 7.508(74) | 9.86(22)
| 9.54(23) |  | 8.91(13) |
$T_{2}$ | 4.750(16) | 6.203(33) | 7.010(68) | 7.787(96)
| 6.687(51) | 8.05(10)⋆⋆ | 8.53(11) | 8.42(14)
| 7.411(81) | 8.25(16) | 8.86(8) | 9.80(17)
| 7.492(68) |  | 8.94(20) |
Table 25: $SU(4)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text). $SU(5)$:
$M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.156(31) | 4.832(40) | 9.04(22) | 10.36(18)
| 5.689(53) | 7.38(11) | 9.48(30) | 9.70(36)
| 7.36(14) | 8.03(30)⋆ |  |
| 7.53(14) |  |  |
$A_{2}$ | 7.31(14) | 9.80(20) | 7.333(59) | 8.05(23)
| 8.85(32) | 9.93(57) | 8.75(19) | 11.03(37)
$E$ | 4.692(22) | 6.152(60) | 8.54(19) | 8.000(51)
| 6.590(62) | 8.19(23) | 9.45(18) | 9.02(22)
| 7.31(10) | 9.51(17) | 9.53(20) | 10.22(13)⋆⋆
| 7.28(10) |  |  |
$T_{1}$ | 7.396(73) | 8.29(13) | 5.915(45) | 7.62(15)
| 7.19(12) | 8.56(15) | 7.018(62) | 8.53(17)⋆⋆
| 8.99(22) | 8.59(16) | 7.624(74) | 9.91(26)
|  |  | 8.36(14) |
$T_{2}$ | 4.686(30) | 6.208(47) | 7.051(72) | 7.87(11)
| 6.621(55) | 7.97(11) | 8.28(15)⋆ | 8.52(9)
| 7.338(84) | 8.45(16) | 8.44(16) | 10.17(9)⋆
| 7.53(10) |  | 9.18(13) |
Table 26: $SU(5)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text). $SU(6)$:
$M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.102(32) | 4.967(43) | 9.37(22) | 10.46(17)
| 6.020(57) | 7.14(13) | 10.58(21) | 10.53(20)⋆⋆
| 7.51(12) | 8.72(31) |  |
| 7.59(13)⋆ |  |  |
$A_{2}$ | 7.46(13) | 9.37(25) | 7.169(96) | 8.71(16)⋆
| 9.58(24)⋆⋆ | 10.48(43) | 9.00(11) | 10.87(38)
|  |  | 9.12(20) |
$E$ | 4.706(30) | 6.098(55) | 8.80(11) | 8.106(50)
| 6.43(11) | 7.93(15) | 8.95(17) | 9.48(33)
| 7.06(13) | 9.58(19) | 9.34(19) | 9.88(11)
| 7.19(17) |  |  |
$T_{1}$ | 7.435(80) | 8.405(88) | 5.847(43) | 7.41(13)
| 7.57(10) | 8.75(11) | 7.066(80) | 8.90(11)
| 9.17(19) | 9.23(11)⋆ | 7.552(85) | 9.86(26)⋆
|  |  | 8.46(11) |
$T_{2}$ | 4.649(28) | 6.157(50) | 6.997(80) | 7.64(14)
| 6.577(71) | 7.92(14) | 8.20(19) | 8.51(9)
| 7.189(72) | 8.25(11)⋆ | 8.864(61) | 10.20(9)
| 7.02(10)⋆ |  | 9.15(11) |
Table 27: $SU(6)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text).

$SU(8)$: $M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.099(26) | 4.755(58) | 8.97(31) | 9.69(35)
| 5.87(7) | 6.90(24) | 10.06(24) | 9.50(39)
| 7.18(13) | 8.25(26) |  |
| 7.61(7) |  |  |
$A_{2}$ | 7.28(14) | 9.80(29) | 7.00(12) | 8.26(28)
| 9.06(34) | 11.28(25) | 8.60(18) | 11.40(23)
|  |  | 8.87(48) |
$E$ | 4.658(32) | 6.091(59) | 8.53(19) | 7.60(11)
| 6.32(7)⋆ | 8.23(13) | 8.94(24)⋆⋆ | 9.18(25)
| 7.26(11) | 9.46(24) | 8.81(35) | 9.57(29)
$T_{1}$ | 7.318(74) | 8.24(15) | 5.801(38) | 7.09(17)
| 7.51(11) | 8.60(27) | 7.089(55) | 8.35(18)
| 9.21(22) | 8.46(22) | 7.43(9)⋆ | 9.60(25)
| 9.21(25) |  | 8.43(12) |
$T_{2}$ | 4.661(21) | 5.995(61) | 6.908(67) | 7.863(66)
| 6.599(84) | 8.00(11) | 8.01(16) | 8.31(12)
| 7.16(8) | 8.38(16) | 8.51(12)⋆ | 9.40(24)
| 7.23(16) |  | 8.92(15) |
Table 28: $SU(8)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text). $SU(10)$:
$M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.102(37) | 4.835(60) | 8.92(37) | 10.04(36)
| 5.99(13) | 7.03(14) | 8.73(60) | 9.79(55)
| 7.15(17) | 8.90(68) |  |
| 7.87(13) |  |  |
$A_{2}$ | 7.20(18) | 8.99(35) | 6.78(15) | 9.06(44)
| 8.35(50) | 11.74(70) | 7.83(44) | 10.70(70)
|  |  | 8.29(103)? |
$E$ | 4.587(27) | 6.12(10) | 8.64(20) | 7.60(14)
| 6.49(9) | 8.03(12) | 9.04(30) | 9.59(33)
| 7.34(12) | 9.36(46) | 9.05(30) | 9.92(22)
$T_{1}$ | 7.14(17) | 8.49(28) | 5.776(41) | 7.27(17)
| 7.84(27) | 8.11(62) | 7.039(62) | 9.00(24)⋆
| 9.32(18)⋆⋆ | 9.53(100)⋆ | 7.79(13) | 10.17(52)
| 9.69(22)⋆⋆ |  | 8.75(21)⋆⋆ |
$T_{2}$ | 4.600(30) | 5.94(8) | 7.071(55) | 7.89(16)
| 6.638(82) | 7.52(17) | 7.60(24) | 7.82(26)⋆
| 7.00(18) | 7.88(30)⋆⋆ | 8.60(13) | 9.26(36)
| 7.19(22) |  | 8.77(28) |
Table 29: $SU(10)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text). $SU(12)$:
$M_{G}/\surd\sigma$ continuum limit
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.151(33) | 4.696(65) | 9.63(25) | 9.57(32)
| 5.914(82) | 6.94(19) | 10.50(51) | 9.90(58)
| 7.07(18) | 9.56(44)⋆ |  |
| 7.61(19) |  |  |
$A_{2}$ | 7.74(22) | 10.16(26) | 6.49(17) | 9.03(22)
| 9.64(26) | 12.09(74) | 8.90(22) | 10.66(60)
|  |  | 9.46(46) |
$E$ | 4.647(33) | 6.22(10) | 8.52(22) | 7.80(14)
| 6.613(80) | 8.28(12)⋆ | 9.56(43) | 9.88(20)
| 7.67(13) |  |  | 10.24(29)
$T_{1}$ | 6.64(27) | 8.59(12) | 5.741(60) | 7.50(17)⋆
| 7.59(21) | 8.95(16) | 7.005(78) | 8.83(14)
| 9.34(18)⋆⋆ | 9.30(30)⋆ | 7.48(14) | 9.51(48)
|  |  | 8.80(23) |
$T_{2}$ | 4.645(33) | 5.97(7) | 6.88(10) | 8.20(13)
| 6.764(57) | 8.05(15) | 8.16(25) | 8.27(31)⋆
| 7.35(15)⋆ | 8.89(22) | 8.48(25) | 10.52(21)⋆⋆
| 7.20(19) |  | 8.21(30) |
Table 30: $SU(12)$ continuum limit of glueball masses in units of the string
tension, for all representations, $R$, of the rotation symmetry of a cube, for
both values of parity, $P$, and charge conjugation, $C$. Ground states and
some excited states. Stars indicate poor fits (see text).

$SU(\infty)$: $M_{G}/\surd\sigma$
---
$R$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$A_{1}$ | 3.072(14) | 4.711(25) | 9.26(16) | 10.10(18)
| 5.805(31)⋆⋆ | 7.050(68) |  | 10.14(23)
| 7.294(63) |  |  |
$A_{2}$ | 7.40(12)† | 9.73(12) | 7.142(75)†⋆ | 8.61(13)⋆
| 9.14(14)⋆ | 11.12(24) | 8.77(10) | 11.38(21)
$E$ | 4.582(14) | 6.108(44) | 8.63(10) | 7.951(53)⋆⋆
| 6.494(33)⋆ | 8.051(60) | 9.14(15) | 9.55(13)
| 7.266(50)⋆ |  |  | 9.84(12)
$T_{1}$ | 7.250(47) | 8.412(76) | 5.760(25) | 7.134(86)
| 7.337(60)⋆ | 8.79(10) | 7.020(39) | 8.65(9)
| 9.142(82) | 9.08(12)⋆⋆ | 7.470(55) | 9.81(17)
|  |  | 8.422(84) |
$T_{2}$ | 4.578(11) | 5.965(28) | 6.957(41) | 7.96(8)†
| 6.579(30) | 7.883(57) | 7.93(11) | 8.22(8)
| 7.121(45) | 8.45(14)† | 8.63(7)$\star$ | 10.26(10)⋆⋆
| 7.122(76) |  |  |
Table 31: Continuum glueball masses in units of the string tension, in the
limit $N\to\infty$. Fits are to $N\geq 2$ or $N\geq 3$ except for values
labelled with a $\dagger$, and $\star$ indicates a poor fit, as explained in
the text. Labels are $R$ for the representations of the rotation symmetry of a
cube, $P$ for parity and $C$ for charge conjugation. continuum $J\sim$ cubic
$R$
---
$J$ |  | cubic $R$
0 | $\sim$ | $A_{1}$
1 | $\sim$ | $T_{1}$
2 | $\sim$ | $E+T_{2}$
3 | $\sim$ | $A_{2}+T_{1}+T_{2}$
4 | $\sim$ | $A_{1}+E+T_{1}+T_{2}$
5 | $\sim$ | $E+2T_{1}+T_{2}$
6 | $\sim$ | $A_{1}+A_{2}+E+T_{1}+2T_{2}$
7 | $\sim$ | $A_{2}+E+2T_{1}+2T_{2}$
8 | $\sim$ | $A_{1}+2E+2T_{1}+2T_{2}$
Table 32: Projection of continuum spin $J$ states onto the cubic
representations $R$. continuum $J^{PC}$ from cubic $R$
---
$J^{PC}$ |  | cubic $R^{PC}$
$0^{++}$gs | $\sim$ |  $A_{1}^{++}$gs
$0^{++}$ex1 | $\sim$ |  $A_{1}^{++}$ex1
$2^{++}$gs | $\sim$ |  $E^{++}$gs + $T_{2}^{++}$gs
$2^{++}$ex1 | $\sim$ |  $E^{++}$ex1 + $T_{2}^{++}$ex1
$3^{++}$gs |  $\sim$* |  $A_{2}^{++}$gs + $T_{1}^{++}$gs(ex1) + $T_{2}^{++}$ex3(ex2)
$4^{++}$gs |  $\sim$* |  $A_{1}^{++}$ex2 + $E^{++}$ex2 +$T_{1}^{++}$ex1(gs) + $T_{2}^{++}$ex2(ex3)
$0^{-+}$gs | $\sim$ |  $A_{1}^{-+}$gs
$0^{-+}$ex1 | $\sim$ |  $A_{1}^{-+}$ex1
$2^{-+}$gs | $\sim$ |  $E^{-+}$gs + $T_{2}^{-+}$gs
$2^{-+}$ex1 | $\sim$ |  $E^{-+}$ex1 + $T_{2}^{-+}$ex1
$1^{-+}$gs | $\sim$ |  $T_{1}^{-+}$gs
$2^{+-}$gs | $\sim$ |  $E^{+-}$gs + $T_{2}^{+-}$ex2
$1^{+-}$gs | $\sim$ |  $T_{1}^{+-}$gs
$1^{+-}$ex1 | $\sim$ |  $T_{1}^{+-}$ex2
$3^{+-}$gs | $\sim$ |  $A_{2}^{+-}$gs + $T_{1}^{+-}$ex1 + $T_{2}^{+-}$gs
$2^{--}$gs | $\sim$ |  $E^{--}$gs + $T_{2}^{--}$gs
$1^{--}$gs | $\sim$ |  $T_{1}^{--}$gs
Table 33: Identification of continuum $J^{PC}$ states from the results for the
cubic representations in Tables 23-31. Ground state denoted by $gs$, $i$’th
excited state by $exi$. Where there is some ambiguity, a single star denotes
’likely’ while two stars indicate ’significant uncertainty’.
$M(J^{PC})/\surd\sigma$ continuum limit
---
$J^{PC}$ | $SU(2)$ | $SU(3)$ | $SU(4)$ | $SU(5)$
$0^{++}$ gs | 3.781(23) | 3.405(21) | 3.271(27) | 3.156(31)
$0^{++}$ ex1 | 6.126(38) | 5.855(41) | 5.827(62) | 5.689(53)
$2^{++}$ gs | 5.349(20) | 4.894(22) | 4.742(15) | 4.690(20)
$2^{++}$ ex1 | 7.22(6)+ | 6.788(40) | 6.694(40) | 6.607(45)
$3^{++}$ gs | 8.13(8)∗ | 7.71(9)∗ |  | 7.29(9)
$4^{++}$ gs |  | 7.60(12)∗ | 7.36(9)∗ | 7.41(10)
$0^{-+}$ gs | 6.017(61) | 5.276(45) | 5.020(46) | 4.832(40)
$0^{-+}$ ex1 | 8.00(15) | 7.29(13) | 7.33(11) | 7.38(11)
$2^{-+}$ gs | 7.017(50) | 6.32(9) | 6.182(33) | 6.187(50)
$2^{-+}$ ex1 | 8.521(65) | 8.18(8) | 7.91(13)∗ | 8.02(11)
$1^{-+}$ gs | 9.06(13) | 8.48(12) | 8.47(11) | 8.29(13)
$2^{+-}$ gs |  | 8.91(15) | 8.84(11) | 8.49(15)
$1^{+-}$ gs |  | 6.065(40) | 5.956(42) | 5.915(45)
$1^{+-}$ ex1 |  | 7.82(6) | 7.51(8) | 7.62(8)
$3^{+-}$ gs |  | 7.27(12) | 7.03(7) | 7.13(7)
$2^{--}$ gs |  | 8.08(15) | 7.79(9) | 7.97(6)
$1^{--}$ gs |  | 8.31(10) | 7.60(9) | 7.62(15)
Table 34: Continuum limit of glueball masses, in units of the string tension,
for those $J^{PC}$ representations we can identify. Ground state denoted by
$gs$, $i$’th excited state by $exi$. Stars explained in text.
$M(J^{PC})/\surd\sigma$ continuum limit
---
$J^{PC}$ | $SU(6)$ | $SU(8)$ | $SU(10)$ | $SU(12)$
$0^{++}$ gs | 3.102(32) | 3.099(26) | 3.102(37) | 3.151(33)
$0^{++}$ ex1 | 6.020(57) | 5.87(7) | 5.99(13) | 5.914(82)
$2^{++}$ gs | 4.678(30) | 4.660(20) | 4.594(25) | 4.646(30)
$2^{++}$ ex1 | 6.54(7) | 6.60(9)∗ | 6.57(9) | 6.71(6)
$3^{++}$ gs | 7.44(8)∗ | 7.34(11)∗ | 7.14(18) |
$4^{++}$ gs |  | 7.20(7)∗ | 7.32(15) |
$0^{-+}$ gs | 4.967(43) | 4.755(58) | 4.835(60) | 4.696(65)
$0^{-+}$ ex1 | 7.14(13) | 6.90(24) | 7.03(14) | 6.94(19)
$2^{-+}$ gs | 6.148(50) | 6.043(55) | 6.01(8) | 6.05(7)
$2^{-+}$ ex1 | 7.92(14) | 8.10(11) | 7.86(12)∗ | 8.16(15)
$1^{-+}$ gs | 8.41(9) | 8.24(15) | 8.49(28) | 8.59(12)
$2^{+-}$ gs | 8.83(9) | 8.52(12) | 8.62(12) | 8.50(17)
$1^{+-}$ gs | 5.847(43) | 5.801(38) | 5.776(41) | 5.741(60)
$1^{+-}$ ex1 | 7.55(9) | 7.43(9)∗ | 7.79(13) | 7.48(14)
$3^{+-}$ gs | 7.067(70) | 6.999(70) | 7.035(50) | 6.89(10)
$2^{--}$ gs | 8.05(5)∗ | 7.78(7) | 7.74(15) | 8.00(15)
$1^{--}$ gs | 7.41(13) | 7.09(17) | 7.27(17) | 7.50(17)
Table 35: Continuum limit of glueball masses, in units of the string tension,
for those $J^{PC}$ representations we can identify. Ground state denoted by
$gs$, $i$’th excited state by $exi$. Stars explained in text.

$M(J^{PC})/\surd\sigma$ lower bounds
---
$1^{++}$ | $3^{-+}$ | $0^{+-}$ | $2^{+-}$ | $0^{--}$ | $3^{--}$
$\geq 9.0$ | $\geq 9.7$ | $\geq 9.2$ | $\geq 8.6$ | $\geq 10.0$ | $\geq 8.6$
Table 36: Lower bounds on masses of some of the low-$J$ ground states not
appearing in Tables 34,35. $M_{G}\,GeV\quad SU(3)$
---
$J$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
0 gs | 1.653(26) | 2.561(40) | $\geq 4.52(15)$ | $\geq 4.81(24)$
0 ex1 | 2.842(40) | 3.54(8) |  |
2 gs | 2.376(32) | 3.07(6) | 4.24(8)* | 3.92(9)
2 ex1 | 3.30(5) | 3.97(7) | $\geq 4.38(13)$ | $\geq 4.55(11)$
1 gs | $\geq 4.52(6)$ | 4.12(8) | 2.944(42) | 4.03(7)
1 ex1 |  | 4.16(8)* | 3.80(6) | $\geq 4.51(9)$
1 ex2 |  | 4.20(9)* |  |
3 gs | 3.74(7)* | $\geq 4.75(13)$ | 3.53(8) | $\geq 4.35(9)$
4 gs | 3.69(8)* | $\geq 4.45(14)$ | 4.38(8)** | $\geq 4.81(24)$
Table 37: Continuum limit of $SU(3)$ glueball masses, in physical $GeV$ units
for those $J^{PC}$ representations we can identify, with lower bounds in those
cases where this is not possible. Ground state denoted by $gs$, first excited
state by $ex$. Stars denote ambiguity. $M_{J}/\surd\sigma$ ; $SU(\infty)$
---
$J$ | P=+,C=+ | P=-,C=+ | P=+,C=- | P=-,C=-
$0$ gs | 3.072(14) | 4.711(26) | $\geq 9.26(16)$ | $\geq 10.10(18)$
$0$ ex | 5.845(50) | 7.050(68) |  |
$2$ gs | 4.599(14) | 6.031(38) | 8.566(76) | 7.910(56)
$2$ ex | 6.582(36) | 7.936(54) | $\geq 9.14(15)$ | $\geq 9.55(13)$
$1$ gs | $\geq 9.14(9)$ | 8.415(76) | 5.760(25) | 7.26(11)
$1$ ex |  |  | 7.473(57) | $\geq 8.65(9)$
$3$ gs | 7.263(56) | $\geq 9.73(12)$ | 6.988(41) | $\geq 8.61(13)$
$4$ gs | 7.182(71) | $\geq 8.79(10)$ | $\geq 9.26(16)$ | $\geq 10.10(18)$
Table 38: Large $N$ extrapolation of continuum glueball masses, in units of
the string tension, for those $J^{PC}$ representations we can identify, with
lower bounds in those cases where this is not possible. Ground state denoted
by $gs$, first excited state by $ex$. $M/\surd\sigma$ ; continuum $SU(3)$
---
paper | $0^{++}$ | $2^{++}$ | $0^{-+}$ | $2^{-+}$ | $1^{+-}$ | $3^{+-}$ | $2^{--}$
this work | 3.405(21) | 4.894(22) | 5.276(45) | 6.32(9) | 6.065(40) | 7.27(12) | 8.08(15)
2005: ref[9] | 3.347(68) | 4.891(65) | 5.11(14) | 6.32(11) | 6.06(15) | 7.43(20) | 8.32(29)
2005: ref[10]⋆ | 3.59(15) | 5.03(11) | 5.39(11) | 6.40(14) | 6.27(11) | 7.58(12) | 8.42(14)
2004: ref[2] | 3.55(7) | 4.78(9) |  |  |  |  |
2001: ref[1] | 3.61(9) | 5.13(22) |  |  |  |  |
Table 39: Some glueball masses in units of the string tension in the continuum
limit of the $SU(3)$ gauge theory: a comparison between this work and some
earlier work. The starred calculation includes our tansformation of the units
from the Sommer scale to the string tension. $M/\surd\sigma$ ; continuum
$SU(8)$
---
paper | $0^{++}$ | $2^{++}$ | $0^{-+}$ | $2^{-+}$ | $1^{+-}$ | $3^{+-}$ | $2^{--}$
this work | 3.099(26) | 4.660(20) | 4.755(58) | 6.043(55) | 5.801(38) | 7.00(7) | 7.78(7)
2005: ref[9] | 3.32(15) | 4.65(19) | 4.72(32) | 5.67(40) | 5.70(29) | 7.74(79) | 7.3(1.4)
2004: ref[2] | 3.55(12) | 4.73(22) |  |  |  |  |
Table 40: Some glueball masses in units of the string tension in the continuum
limit of the $SU(8)$ gauge theory: a comparison between this work and some
earlier work. $Q_{L}(n_{c})$ : $SU(8)$, $Q=2$
---
| $\beta=45.50,12^{3}20$ | $\beta=46.70,16^{3}24$ | $\beta=47.75,20^{3}30$
$n_{c}$ | $\bar{Q}_{L}$ | $\sigma_{Q_{L}}$ | $\bar{Q}_{L}$ | $\sigma_{Q_{L}}$ | $\bar{Q}_{L}$ | $\sigma_{Q_{L}}$
0 | 0.275(21) | 0.836 | 0.333(23) | 1.282 | 0.410(35) | 1.956
1 | 1.199(6) | 0.202 | 1.286(5) | 0.265 | 1.345(7) | 0.351
2 | 1.575(3) | 0.070 | 1.631(2) | 0.082 | 1.665(3) | 0.103
3 | 1.699(2) | 0.041 | 1.7465(7) | 0.044 | 1.7743(9) | 0.053
4 | 1.7576(7) | 0.030 | 1.8005(6) | 0.030 | 1.8254(6) | 0.035
8 | 1.8437(3) | 0.016 | 1.8781(3) | 0.014 | 1.8970(2) | 0.014
12 | 1.8744(3) | 0.013 | 1.9046(2) | 0.010 | 1.9212(1) | 0.009
16 | 1.8912(3) | 0.011 | 1.9186(2) | 0.008 | 1.9339(1) | 0.007
20 | 1.9021(3) | 0.010 | 1.9275(1) | 0.007 | 1.9418(1) | 0.006
Table 41: Lattice topological charge $Q_{L}$ as function of number of cooling
sweeps, $n_{c}$, for fields which have $Q_{L}\simeq 2$ after 20 cooling
sweeps: the average value, with error, and standard deviation. For small,
intermediate and large $\beta$, in $SU(8)$, all with similar volumes in
physical units. topology tunnelling time
---
$\beta$ | $\tau_{Q}$ | $\tilde{\tau}_{Q}$ | $\beta$ | $\tau_{Q}$ | $\tilde{\tau}_{Q}$
$SU(3)$ | $SU(4)$
6.235 | 100(2) | 112(2) | 11.20 | 499(9) | 520(9)
6.338 | 247(6) | 283(6) | 11.40 | 2173(42) | 2275(45)
6.50 | 1056(44) | 1236(50) | 11.60 | 12674(636) | 13371(690)
6.60 | 3399(188) | 3731(190) |  |  |
6.70 | 7973(674) | 8973(744) |  |  |
$SU(5)$ | $SU(6)$
17.22 | 189(4) | 196(4) | 25.05 | 1158(31) | 1177(31)
17.43 | 695(13) | 721(13) | 25.32 | 5992(262) | 6105(272)
17.63 | 2686(90) | 2749(93) | 25.55 | $23.6(1.8)\times 10^{3}$ | $24.4(1.8)\times 10^{3}$
18.04 | $35.7(3.7)\times 10^{3}$ | $37.0(3.7)\times 10^{3}$ | 26.22 | $1.43(99)\times 10^{6}$ | $1.43(99)\times 10^{6}$
18.375 | $40(17)\times 10^{4}$ | $44(19)\times 10^{4}$ |  |  |
$SU(8)$ |
44.10 | 514(10) | 528(10) |  |  |
44.85 | $26.3(3.5)\times 10^{3}$ | $26.3(3.5)\times 10^{3}$ |  |  |
45.50 | $7.1(3.5)\times 10^{5}$ | $7.1(3.5)\times 10^{5}$ |  |  |
45.50 | $12.5(5.1)\times 10^{5}$ | $12.5(5.1)\times 10^{5}$ |  |  |
$SU(10)$ | $SU(12)$
69.20 | 4793(211) | 4831(214) | 99.86 | $4.83(58)\times 10^{4}$ | $4.83(58)\times 10^{4}$
Table 42: Average number of sweeps, $\tau_{Q}$, between $\Delta Q=\pm 1$
changes normalised to a standard space-time volume of $(3/\surd\sigma)^{4}$,
with $\tilde{\tau}_{Q}$ including a correction for ‘near-dislocations’.
$\ln\\{\tau_{Q}\\}=b-c(N)\ln\\{a\surd\sigma\\}$
---
$N$ | $c$ | $11N/3-5$
3 | 6.82(7) | 6.0
4 | 9.27(12) | 9.6$\dot{6}$
5 | 11.13(20) | 13.3$\dot{3}$
6 | 13.50(30) | 17.0
8 | 17.58(43) | 24.3$\dot{3}$
Table 43: Fitted values of $c(N)$ from Fig.27 compared to the asymptotic
dilute gas prediction in eqn(34). $SU(2)$ topology
---
$\beta$ | lattice | $\langle Q_{L}^{2}\rangle$ | $\langle Q_{I}^{2}\rangle$
2.2986 | $8^{3}16$ | 2.266(9) | 3.057(12)
| $12^{3}16$ | 7.694(42) | 9.600(49)
2.3714 | $10^{3}16$ | 2.705(9) | 3.544(12)
| $14^{3}16$ | 7.343(61) | 9.220(77)
2.427 | $12^{3}16$ | 2.824(7) | 3.551(8)
| $16^{4}$ | 6.635(56) | 8.066(64)
| $20^{3}16$ | 13.09(19) | 15.41(22)
| $24^{3}16$ | 22.56(20) | 26.18(23)
2.509 | $16^{3}20$ | 3.411(13) | 4.067(16)
| $22^{3}20$ | 8.822(58) | 10.137(65)
2.60 | $22^{3}30$ | 4.491(31) | 5.109(33)
| $30^{4}$ | 11.48(10) | 12.65(12)
2.65 | $26^{3}34$ | 4.702(41) | 5.268(45)
2.70 | $28^{3}40$ | 3.786(36) | 4.181(39)
| $40^{4}$ | 11.07(14) | 12.09(14)
2.75 | $34^{3}46$ | 4.328(88) | 4.711(94)
2.80 | $40^{3}54$ | 4.43(12) | 4.76(12)
Table 44: Average values of $Q^{2}_{L}$, after 20 cooling sweeps, in $SU(2)$ for various values of $\beta$ and lattice sizes. $Q_{I}$ is the projection of $Q_{L}$ to an integer value. $SU(3)$ topology |  $SU(4)$ topology
---|---
$\beta$ | lattice | $\langle Q^{2}_{L}\rangle$ | $\langle Q_{I}^{2}\rangle$ | $\beta$ | lattice | $\langle Q^{2}_{L}\rangle$ | $\langle Q_{I}^{2}\rangle$
5.6924 | $8^{3}16$ | 4.034(21) | 5.452(29) | 10.70 | $12^{3}16$ | 5.813(37) | 7.084(48)
5.80 | $10^{3}16$ | 4.021(13) | 5.111(16) | 10.85 | $14^{3}20$ | 5.718(45) | 6.674(53)
5.8941 | $12^{3}16$ | 3.570(35) | 4.352(43) | 11.02 | $18^{3}20$ | 6.010(95) | 6.78(11)
5.99 | $14^{3}20$ | 3.790(48) | 4.428(56) | 11.20 | $22^{4}$ | 5.97(33) | 6.57(36)
6.0625 | $14^{3}20$ | 2.312(22) | 2.649(26) | 11.40 | $26^{4}$ | 6.19(67) | 6.68(73)
6.235 | $18^{3}26$ | 2.32(12) | 2.55(13) | 11.60 | $30^{4}$ | 5.52(73) | 5.80(78)
6.3380 | $22^{3}30$ | 2.72(13) | 2.94(14) |  |  |  |
6.50 | $26^{3}38$ | 2.09(20) | 2.23(21) |  |  |  |
6.60 | $32^{3}40$ | 2.60(35) | 2.74(37) |  |  |  |
6.70 | $36^{3}44$ | 1.54(30) | 1.61(31) |  |  |  |
Table 45: Average values of $Q_{L}^{2}$, after 20 cooling sweeps, in $SU(3)$ and in $SU(4)$ for the values of $\beta$ and lattices shown. $Q_{I}$ is the projection of $Q_{L}$ to an integer value. $SU(5)$ topology |  $SU(6)$ topology
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$\beta$ | lattice | $\langle Q^{2}_{L}\rangle$ | $\langle Q_{I}^{2}\rangle$ | $\beta$ | lattice | $\langle Q^{2}_{L}\rangle$ | $\langle Q_{I}^{2}\rangle$