Can gluon spin contribute to that of nucleon?
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an Gluon Spin Contribute to That of Nucleon?1, 2 O. V. Teryaev Joint Institute for Nuclear Research, Dubna, 141980 Russia Abstract—The transformation of angular momentum to its Belinfante form requires the smooth behav iour of classical fields at infinity which for the case of quantum operators transforms to the smooth behav iour of matrix elements at small momentum transfers. For the case of quarks this provides the kinematical counterpart of UA(1) problem while for gluons there is a contradiction between kinematics and dynamics governed by KogutSusskind pole. This may result in the violation of Equivalence Principle for nucleons or in the stringent constraints to the strange quark polarization in nucleons, while the most likely out come would be the impossibility to separate gluon angular momentum to spin and orbital parts in the meaningful way. DOI: 10.1134/S1063779614011048 21
1. INTRODUCTION
This relation in QCD is attributed to supersymmetry, while it is possible to derive the similar relation for scalar gluons constraining the only relevant kernel (~1 – x)
The possibility to separate gluon angular momen tum contribution to spin and orbital parts in cur rently under the intensive scrutiny [1]. Here I address the particular problem emerging when angular momentum us transformed to its Belinfante form. The crucial role is played by the smooth behaviour for large distances and small momenta of the matrix element of quark axial current and gluon topological current.
1
∫ dx ( 1 – 3x )P
∫ 0
M
μ, νρ
ν μρ ρ μν 1 μνρσ 5 = ⑀ J Sσ + x T – x T . 2
(2)
Performing the transformation to the symmetric Belinfante energy–momentum tensor one has M
μ, νρ
ν
μρ
μν
ρ
= x TB – x TB .
(3)
As the conservation of the angular momentum imme diately leads to the symmetry of T μρ, the latter implies that ⑀ μνρα M
μ, νρ
= 0.
(4)
One should conclude that the totally antisymmetric quark spin tensor is somehow cancelled and doesn’t contribute to the total angular momentum [5]. How ever, it is still possible [3] to get the constraints for the surface terms of classical fields
1
∫
(1)
While RI for classical fields requires their decrease at infinity, it puts the constraints for the behaviour of the matrix elements of respective operators at low momentum transfers, which are of special interest in the nonperturbative case. Let us start with the quark spin extracting it from the total angular momentum while the gluon angular momentum is (traditionally!) not decomposed to spin and orbital parts contained together with the quark orbital momentum in the con tribution of EMT
The Relocalization (Belinfante) Invariance (RI) is providing the possibility to perform a transformation of the densities of conserved charges and represent the total angular momentum in an “orbital” form with Belinfante symmetrized energy momentum tensor (EMT). Let us stress that it is this tensor which describes the coupling to gravity and enters the rele vant gravitational formfactors. The matching of RI with quantum theory happens not to be
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