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Register a new userGauge dependence in the anomalous dimension of the gauge invariant canonical decomposition for proton momentum
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The gauge dependence in the anomalous dimension of the gauge-invariant-canonical-energy-momentum tensor for proton is studied by the background field method. The naive calculation shows the problem, the absence of the counter term in the gluonic sectors. The analysis shows that the result [Chen et al., Phys. Rev. Lett. 103, 062001 (2009)] is derived from the background field method after we introduced a trick to avoid the problem except for the gluon-to-gluon sector; it is gauge dependent. The possible reason of this gauge-dependent result comes from the nontrivial treatment of the condition $F^{mu u}_{pure}=0$ at a higher order. This result shows that one needs a further improvement in treating this condition with a covariant way at a higher order by the background field method. In particular, we have to focus on two checkpoints, the gauge independence and zero eigenvalue in the anomalous-dimension matrix, in order to test the validity of the gauge-invariant-canonical-energy-momentum tensor.
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The anomalous dimension for the gauge-invariant-canonical decomposition of the energy-momentum tensor for quarks and gluons is studied by the background field method. In particular, the consistency between the background field method and the renormalization in the gluonic sectors is investigated. The analysis shows that the naive gauge-invariant-decomposition has an inconsistency between its definition and the renormalization in the background field method. Although we try to consider a trick to overcome this inconsistency in computing the anomalous dimension, the gauge-parameter dependence remains in the final result. This result should be extended to the problems on the gauge-invariant-canonical-spin decomposition.
Abelian topologically massive gauge theories (TMGT) provide a topological mechanism to generate mass for a bosonic p-tensor field in any spacetime dimension. These theories include the 2+1 dimensional Maxwell-Chern-Simons and 3+1 dimensional Cremmer-Scherk actions as particular cases. Within the Hamiltonian formulation, the embedded topological field theory (TFT) sector related to the topological mass term is not manifest in the original phase space. However through an appropriate canonical transformation, a gauge invariant factorisation of phase space into two orthogonal sectors is feasible. The first of these sectors includes canonically conjugate gauge invariant variables with free massive excitations. The second sector, which decouples from the total Hamiltonian, is equivalent to the phase space description of the associated non dynamical pure TFT. Within canonical quantisation, a likewise factorisation of quantum states thus arises for the full spectrum of TMGT in any dimension. This new factorisation scheme also enables a definition of the usual projection from TMGT onto topological quantum field theories in a most natural and transparent way. None of these results rely on any gauge fixing procedure whatsoever.
Is gauge-invariant complete decomposition of the nucleon spin possible? Although it is a difficult theoretical question which has not reached a complete consensus yet, a general agreement now is that there are at least two physically inequivalent gauge-invariant decompositions (I) and (II) of the nucleon. %The one is a nontrivial gauge-invariant %generalization of the Jaffe-Manohar decomposition. %The other is an extension of the Ji decomposition, which allows %a gauge-invariant decomposition of the total gluon angular %momentum into the intrinsic spin and orbital parts. In these two decompositions, the intrinsic spin parts of quarks and gluons are just common. What discriminate these two decompositions are the orbital angular momentum parts. The orbital angular momenta of quarks and gluons appearing in the decomposition (I) are the so-called mechanical orbital angular momenta, while those appearing in the decomposition (II) are the generalized (gauge-invariant) canonical ones. By this reason, these decompositions are also called the mechanical and canonical decompositions of the nucleon spin, respectively. A crucially important question is which decomposition is more favorable from the observational viewpoint. The main objective of this concise review is to try to answer this question with careful consideration of recent intensive researches on this problem.
The propagator of a gauge boson, like the massless photon or the massive vector bosons $W^pm$ and $Z$ of the electroweak theory, can be derived in two different ways, namely via Greens functions (semi-classical approach) or via the vacuum expectation value of the time-ordered product of the field operators (field theoretical approach). Comparing the semi-classical with the field theoretical approach, the central tensorial object can be defined as the gauge boson projector, directly related to the completeness relation for the complete set of polarisation four-vectors. In this paper we explain the relation for this projector to different cases of the $R_xi$ gauge and explain why the unitary gauge is the default gauge for massive gauge bosons.
The question whether the total gluon angular momentum in the nucleon can be decomposed into its spin and orbital parts without conflict with the gauge-invariance principle has been an object of long-lasting debate. Despite a remarkable progress achieved through the recent intensive researches, the following two issues still remains to be clarified more transparently. The first issue is to resolve the apparent conflict between the proposed gauge-invariant decomposition of the total gluon angular momentum and the textbook statement that the total angular momentum of the photon cannot be gauge-invariantly decomposed into its spin and orbital parts. We show that this problem is also inseparably connected with the uniqueness or non-uniqueness problem of the nucleon spin decomposition. The second practically more important issue is that, among the two physically inequivalent decompositions of the nucleon spin, i.e. the canonical type decomposition and the mechanical type decomposition, which can we say is more physical or closer to direct observation ? In the present paper, we try to answer both these questions as clearly as possible.
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