Abstract
We study the effect of Lorentz symmetry violation (LSV) on the behavior at high energy of $SU(N)$ gauge theory with quarks in the fundamental representation. The approach is similar to that for QED treated in a previous paper. In contrast to QED, standard Lorentz invariant QCD is asymptotically free. Our aim is to explore the structure of the renormalization group at high energy and hence weak coupling without requiring the Lorentz symmetry breaking to be small. The simplest type of LSV leaves the theory invariant under a subgroup of the Lorentz group that preserves a (timelike) 4-vector. We examine this case in detail and find that asymptotic freedom is frustrated. That is, at sufficiently high energy, the running coupling constant attains a minimum value before increasing again, while the LSV parameter increases without bound.
Highlights
Lorentz symmetry violation (LSV) in QED has been studied by a number of authors concerned with its consistency with causality, unitarity [1,2,3], the structure of asymptotic states, and renormalization theory [4,5,6]
In previous papers [7,8], we studied some of these issues in QED starting with a premetric formulation [9,10] based on an action d4 xUμνστ Fμν ðxÞFστ ðxÞ; ð1Þ
We look in detail only at the simplest type of LSV, we set out the general theory in a manner parallel to Ref. [7] in order to clarify the logical structure of the argument
Summary
Lorentz symmetry violation (LSV) in QED has been studied by a number of authors concerned with its consistency with causality, unitarity [1,2,3], the structure of asymptotic states, and renormalization theory [4,5,6]. This change of coordinates involves UV divergences through the perturbative expansion in the coupling constant The structure of this argument is completely evident in the application of the theory to the case of Petrov class O. As in the case of QED [7], the quark field can engender LSV in the model through its contribution to vacuum polarization provided the associated metric for quark propagation shares with the gluon metric an invariance under a subgroup of the Lorentz group that is the little group of the given 4-vector [22].
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