5 QCD on the Lattice
203
order of magnitude. This is not observed in experiment, where the lightest scalar
mesons are found to lie 600–700 MeV above the pseudoscalar octet. One therefore
concludes that the symmetry must be spontaneously broken. The term “spontaneous
breaking” refers to the fact that theories like QCD possess more internal symmetries
than those that can be inferred from the particle spectrum. In general, spontaneously
broken symmetries are not realized as symmetry transformations involving the
physical states of the theory. In particular, the ground state, i.e. the vacuum, is not
invariant under the transformation. As discussed in many textbooks, it is precisely
the invariance of the vacuum under the symmetry transformation that is required
to ensure the degeneracy of the particle spectrum. If the vacuum is not invariant,
certain operators may acquire a non-vanishing expectation value. In fact, a sufficient
condition for the spontaneous breaking of the physical SU(3) L ⊗ SU (3) R chiral
symmetry is fulfilled if the expectation value of the scalar density, ¯
, is non-zero,
i.e.
¯
≡
¯
uu + ¯
dd + ¯
ss
= 0.
(5.168)
Furthermore, according to Goldstone’s theorem [101], the generator of each broken
symmetry is associated with a massless particle. Since the masses of the members
of the pseudoscalar octet are rather small in comparison with the proton mass, they
are identified as the Goldstone bosons of the spontaneously broken chiral symmetry.
Spontaneous chiral symmetry breaking is an entirely non-perturbative phenomenon. The task is then to explore the breaking mechanism and compute the
value of the quark condensate
¯
. As shall be outlined below, this can be achieved
through the interplay of lattice simulations and effective low-energy descriptions of
QCD.
5.6.1 Chiral Perturbation Theory
Chiral Perturbation Theory (ChPT) has already been mentioned in connection with
extrapolations of lattice data to the physical values of the up- and down-quark
masses, and also in the context of lattice determinations of the strange quark mass.
Here we present a brief introduction into the general formalism. More thorough
reviews can be found in Refs. [102, 103].
Chiral Perturbation Theory is an effective theory, based on a systematic expansion of the low-energy dynamics of QCD in powers of the 4-momentum and the
quark mass about the chiral limit [89, 90], i.e.
L eff = L
(2)
eff + L
(4)
eff + . . . ,
(5.169)
where the superscripts label the order of the expansion in powers of p. In contrast to
QCD, the basic degrees of freedom which appear in L eff are the Goldstone bosons,
203
order of magnitude. This is not observed in experiment, where the lightest scalar
mesons are found to lie 600–700 MeV above the pseudoscalar octet. One therefore
concludes that the symmetry must be spontaneously broken. The term “spontaneous
breaking” refers to the fact that theories like QCD possess more internal symmetries
than those that can be inferred from the particle spectrum. In general, spontaneously
broken symmetries are not realized as symmetry transformations involving the
physical states of the theory. In particular, the ground state, i.e. the vacuum, is not
invariant under the transformation. As discussed in many textbooks, it is precisely
the invariance of the vacuum under the symmetry transformation that is required
to ensure the degeneracy of the particle spectrum. If the vacuum is not invariant,
certain operators may acquire a non-vanishing expectation value. In fact, a sufficient
condition for the spontaneous breaking of the physical SU(3) L ⊗ SU (3) R chiral
symmetry is fulfilled if the expectation value of the scalar density, ¯
, is non-zero,
i.e.
¯
≡
¯
uu + ¯
dd + ¯
ss
= 0.
(5.168)
Furthermore, according to Goldstone’s theorem [101], the generator of each broken
symmetry is associated with a massless particle. Since the masses of the members
of the pseudoscalar octet are rather small in comparison with the proton mass, they
are identified as the Goldstone bosons of the spontaneously broken chiral symmetry.
Spontaneous chiral symmetry breaking is an entirely non-perturbative phenomenon. The task is then to explore the breaking mechanism and compute the
value of the quark condensate
¯
. As shall be outlined below, this can be achieved
through the interplay of lattice simulations and effective low-energy descriptions of
QCD.
5.6.1 Chiral Perturbation Theory
Chiral Perturbation Theory (ChPT) has already been mentioned in connection with
extrapolations of lattice data to the physical values of the up- and down-quark
masses, and also in the context of lattice determinations of the strange quark mass.
Here we present a brief introduction into the general formalism. More thorough
reviews can be found in Refs. [102, 103].
Chiral Perturbation Theory is an effective theory, based on a systematic expansion of the low-energy dynamics of QCD in powers of the 4-momentum and the
quark mass about the chiral limit [89, 90], i.e.
L eff = L
(2)
eff + L
(4)
eff + . . . ,
(5.169)
where the superscripts label the order of the expansion in powers of p. In contrast to
QCD, the basic degrees of freedom which appear in L eff are the Goldstone bosons,