5 QCD on the Lattice
173
shows that the data obtained for N f = 2 are closer to the experimental results in
the continuum limit in comparison with their quenched counterparts. However, the
figure also shows that the extrapolation of unquenched data is not well constrained,
since only three data points are available. Clearly, additional simulations at smaller
lattice spacings and quark masses are required for a solid determination of the total
error in unquenched calculations of the light hadron spectrum.
It should also be noted that the various discretizations of the quark action
have complementary advantages and shortcomings. While simulations with Wilson
quarks have in the past been restricted to quark masses not much smaller than half
the strange quark mass for algorithmic reasons, the use of staggered fermions in
conjunction with the rooting procedure may be afflicted with conceptual problems
(see the discussion in Sect. 5.2.6). Domain wall and overlap fermions are per se
more expensive to simulate. In simulations based on tmQCD the incorporation of a
third, heavier quark flavour is quite complicated. Thus, progress in this area is likely
to be made through the combined information from different discretizations.
5.3.2 Glueballs
In addition to bound states composed of a quark-antiquark pair or, alternatively,
three quarks, QCD is also widely believed to support the existence of glueballs, i.e.
bound states consisting mainly of gluonic degrees of freedom. Although several
candidates for such states have been proposed (e.g. the f 0 (1370), f 0 (1500) and
f 0 (1710)), the experimental difficulty consists in their unambiguous identification
as glueballs. To this end, they need to be distinguished from “conventional” flavoursinglet meson resonances in the scalar channel. Predictions for the masses and
widths of glueballs from lattice QCD provide crucial input for this task.
The basic principles of mass calculations for glueballs in lattice QCD are the
same as for bound states composed of quark degrees of freedom: first one must
define an interpolating operator with the appropriate quantum numbers of the
glueball state in question. That is, the operator must transform correctly under
spin, parity and charge conjugation. At this point a complication arises: the lattice
breaks all continuous space-time symmetries, such that Lorentz-invariance or—in
the language of Euclidean field theory—rotational invariance is only recovered in
the continuum limit. At non-zero lattice spacing the spin assignment is therefore
ambiguous. Since the gluon field is represented by link variables, any glueball
operator must be constructed from particular combinations of Wilson loops, i.e.
products of link variables along closed paths on a hypercubic lattice (see Fig. 5.7).
Operators constructed in this way transform under irreducible representations
(IRs) of the octahedral group O h , which are conventionally labelled A 1 , A 2 , E, T 1
and T 2 . By computing the relations between the IRs of O h and SU(2) one finds
that each IR in the set {A 1 , A 2 , E, T 1 , T 2 } corresponds to infinitely many spins in
the continuum. For instance, A 1 transforms not only like a scalar (spin 0) state, but
also contributes to spin 4 and yet higher spin states. Similarly, the lowest states to
173
shows that the data obtained for N f = 2 are closer to the experimental results in
the continuum limit in comparison with their quenched counterparts. However, the
figure also shows that the extrapolation of unquenched data is not well constrained,
since only three data points are available. Clearly, additional simulations at smaller
lattice spacings and quark masses are required for a solid determination of the total
error in unquenched calculations of the light hadron spectrum.
It should also be noted that the various discretizations of the quark action
have complementary advantages and shortcomings. While simulations with Wilson
quarks have in the past been restricted to quark masses not much smaller than half
the strange quark mass for algorithmic reasons, the use of staggered fermions in
conjunction with the rooting procedure may be afflicted with conceptual problems
(see the discussion in Sect. 5.2.6). Domain wall and overlap fermions are per se
more expensive to simulate. In simulations based on tmQCD the incorporation of a
third, heavier quark flavour is quite complicated. Thus, progress in this area is likely
to be made through the combined information from different discretizations.
5.3.2 Glueballs
In addition to bound states composed of a quark-antiquark pair or, alternatively,
three quarks, QCD is also widely believed to support the existence of glueballs, i.e.
bound states consisting mainly of gluonic degrees of freedom. Although several
candidates for such states have been proposed (e.g. the f 0 (1370), f 0 (1500) and
f 0 (1710)), the experimental difficulty consists in their unambiguous identification
as glueballs. To this end, they need to be distinguished from “conventional” flavoursinglet meson resonances in the scalar channel. Predictions for the masses and
widths of glueballs from lattice QCD provide crucial input for this task.
The basic principles of mass calculations for glueballs in lattice QCD are the
same as for bound states composed of quark degrees of freedom: first one must
define an interpolating operator with the appropriate quantum numbers of the
glueball state in question. That is, the operator must transform correctly under
spin, parity and charge conjugation. At this point a complication arises: the lattice
breaks all continuous space-time symmetries, such that Lorentz-invariance or—in
the language of Euclidean field theory—rotational invariance is only recovered in
the continuum limit. At non-zero lattice spacing the spin assignment is therefore
ambiguous. Since the gluon field is represented by link variables, any glueball
operator must be constructed from particular combinations of Wilson loops, i.e.
products of link variables along closed paths on a hypercubic lattice (see Fig. 5.7).
Operators constructed in this way transform under irreducible representations
(IRs) of the octahedral group O h , which are conventionally labelled A 1 , A 2 , E, T 1
and T 2 . By computing the relations between the IRs of O h and SU(2) one finds
that each IR in the set {A 1 , A 2 , E, T 1 , T 2 } corresponds to infinitely many spins in
the continuum. For instance, A 1 transforms not only like a scalar (spin 0) state, but
also contributes to spin 4 and yet higher spin states. Similarly, the lowest states to