4 QCD: The Theory of Strong Interactions
89
gauge fields are physical because of the constraints from gauge invariance which
can be used to eliminate the dependent variables. This is already true for abelian
theories and we are familiar with the QED case. One introduces a gauge fixing term
(an additional term in the lagrangian density that acts as a Lagrange multiplier in
the action extremisation). One can choose to preserve manifest Lorentz invariance.
In this case, one adopts a covariant gauge, like the Lorentz gauge, and in QED
one proceeds according to the formalism of Gupta-Bleuler. Or one can give up
explicit formal covariance and work in a non covariant gauge, like the Coulomb
or the axial gauges, and only quantise the physical degrees of freedom (in QED the
transverse components of the photon field). While this is all for an abelian gauge
theory, in the non-abelian case some additional complications arise, in particular
the necessity to introduce ghosts for the formulation of Feynman rules. As we
have seen, there are in general as many ghost fields as gauge bosons and they
appear in the form of a transformation Jacobian in the Feynman diagram functional
integral. Ghosts only propagate in closed loops and their vertices with gluons can be
included as additional terms in the lagrangian density which are fixed once the gauge
fixing terms and their infinitesimal gauge transformations are specified. Finally
the complete Feynman rules in a given gauge can be obtained and they appear in
Fig. 4.1.
Once the Feynman rules are derived we have a formal perturbative expansion
but loop diagrams generate infinities. First a regularisation must be introduced,
compatible with gauge symmetry and Lorentz invariance. This is possible in QCD.
In principle one can introduce a cut-off K (with dimensions of energy), for example,
a’ la Pauli-Villars. But at present the universally adopted regularisation procedure is
dimensional regularisation that we will briefly describe later on. After regularisation
the next step is renormalisation. In a renormalisable theory (like for all gauge
theories in four spacetime dimensions and for QCD in particular) the dependence
on the cutoff can be completely reabsorbed in a redefinition of particle masses,
of gauge coupling(s) and of wave function normalisations. After renormalisation
is achieved the perturbative definition of the quantum theory that corresponds to
a classical lagrangian like in Eq. (4.1) is completed. In the QCD Lagrangian of
Eq. (4.1) quark masses are the only parameters with physical dimensions (we work
in the natural system of units ¯
h = c = 1). Naively we would expect that massless
QCD is scale invariant. This is actually true at the classical level. Scale invariance
implies that dimensionless observables should not depend on the absolute scale of
energy but only on ratios of energy-dimensional variables. The massless limit should
be relevant for the asymptotic large energy limit of processes which are non singular
for m → 0.
The naive expectation that massless QCD should be scale invariant is false in
the quantum theory. The scale symmetry of the classical theory is unavoidably
destroyed by the regularisation and renormalisation procedure which introduce a
dimensional parameter in the quantum version of the theory. When a symmetry
of the classical theory is necessarily destroyed by quantisation, regularisation and
renormalisation one talks of an “anomaly”. So, in this sense, scale invariance in
massless QCD is anomalous.
89
gauge fields are physical because of the constraints from gauge invariance which
can be used to eliminate the dependent variables. This is already true for abelian
theories and we are familiar with the QED case. One introduces a gauge fixing term
(an additional term in the lagrangian density that acts as a Lagrange multiplier in
the action extremisation). One can choose to preserve manifest Lorentz invariance.
In this case, one adopts a covariant gauge, like the Lorentz gauge, and in QED
one proceeds according to the formalism of Gupta-Bleuler. Or one can give up
explicit formal covariance and work in a non covariant gauge, like the Coulomb
or the axial gauges, and only quantise the physical degrees of freedom (in QED the
transverse components of the photon field). While this is all for an abelian gauge
theory, in the non-abelian case some additional complications arise, in particular
the necessity to introduce ghosts for the formulation of Feynman rules. As we
have seen, there are in general as many ghost fields as gauge bosons and they
appear in the form of a transformation Jacobian in the Feynman diagram functional
integral. Ghosts only propagate in closed loops and their vertices with gluons can be
included as additional terms in the lagrangian density which are fixed once the gauge
fixing terms and their infinitesimal gauge transformations are specified. Finally
the complete Feynman rules in a given gauge can be obtained and they appear in
Fig. 4.1.
Once the Feynman rules are derived we have a formal perturbative expansion
but loop diagrams generate infinities. First a regularisation must be introduced,
compatible with gauge symmetry and Lorentz invariance. This is possible in QCD.
In principle one can introduce a cut-off K (with dimensions of energy), for example,
a’ la Pauli-Villars. But at present the universally adopted regularisation procedure is
dimensional regularisation that we will briefly describe later on. After regularisation
the next step is renormalisation. In a renormalisable theory (like for all gauge
theories in four spacetime dimensions and for QCD in particular) the dependence
on the cutoff can be completely reabsorbed in a redefinition of particle masses,
of gauge coupling(s) and of wave function normalisations. After renormalisation
is achieved the perturbative definition of the quantum theory that corresponds to
a classical lagrangian like in Eq. (4.1) is completed. In the QCD Lagrangian of
Eq. (4.1) quark masses are the only parameters with physical dimensions (we work
in the natural system of units ¯
h = c = 1). Naively we would expect that massless
QCD is scale invariant. This is actually true at the classical level. Scale invariance
implies that dimensionless observables should not depend on the absolute scale of
energy but only on ratios of energy-dimensional variables. The massless limit should
be relevant for the asymptotic large energy limit of processes which are non singular
for m → 0.
The naive expectation that massless QCD should be scale invariant is false in
the quantum theory. The scale symmetry of the classical theory is unavoidably
destroyed by the regularisation and renormalisation procedure which introduce a
dimensional parameter in the quantum version of the theory. When a symmetry
of the classical theory is necessarily destroyed by quantisation, regularisation and
renormalisation one talks of an “anomaly”. So, in this sense, scale invariance in
massless QCD is anomalous.