Lorentz Symmetry and the Internal Structure of the Nucleon

To investigate the internal structure of the nucleon, it is useful to introduce quantities that do not transform properly under Lorentz symmetry, such as the four-momentum of the quarks in the nucleon, the amount of the nucleon spin contributed by quark spin, etc. In this paper, we discuss to what extent these quantities do provide Lorentz-invariant descriptions of the nucleon structure. Typeset using REVTEX 1 In field theory, one often encounters various densities consisting of elementary fields, space-time coordinates and their derivatives: baryon current, momentum density, angular momentum density, etc. These densities are Lorentz covariant, i.e., under Lorentz tranformations, they transform properly as four-vectors or four-tensors. In many cases, we are interested also in the charges defined from these densities. Considering a generic density j, one can define a charge according to, Q = ∫ dxj . (1) Generally speaking, Q no longer transforms properly under Lorentz transformations. The condition for Q to be Lorentz covariant is well known [1]: The density j must be conserved relative to the index μ, ∂μj μα... = 0 . (2) Indeed in most of the applications, one considers charges from conserved densities. Nevertheless, it is useful to consider charges defined from non-conserved densities. For instance, in quantum chromodynamics (QCD), the energy-momentum density consists of the sum of quark and gluon contributions, T μν = T μν q + T μν g . (3) The total density is conserved due to translational invariance, ∂μT μν = 0. Therefore, the total momentum operator P , P μ = ∫ dxT 0μ , (4) tranforms like a four-vector under Lorentz symmetry. Meanwhile, one can also introduce the notion of the four-momenta carried separately by quarks and gluons, P μ q,g(μ) = ∫ dxT 0μ q,g , (5) where the non-conservation of T 0μ q,g calls for a renormalization scale μ. It is quite obvious that P μ q,g do not transform like four-vectors, and therefore significance of such quantities appears doubtful. However, the exact transformation property of the expectation values of P μ q,g is simple to derive. The forward matrix element of the total energy-momentum density in a nucleon state is, 〈p|T μν |p〉 = 2pp . (6) Here the covariant normalization of the nucleon state is used. From the above, one can easily obtain the usual matrix element of the momentum operator. The matrix elements of quark and gluon parts of the density involve two Lorentz structures, 〈p|T μν q,g |p〉 = 2Aq,g(μ)pp + 2Bq,g(μ)g (7) where Aq,g and Bq,g are scalar constants. Comparing with Eq. (6), one has the following constraints, 2 Aq(μ) + Ag(μ) = 1 , Bq(μ) +Bg(μ) = 0 . (8) Moreover, the quark and gluon contributions to the nucleon four-momentum are, 〈p|P μ q,g|p〉 = Aq,g(μ)p +Bq,g(μ)g/(2p) . (9) The above equation defines transformation properties of the expectation values of P μ q,g. The presence of the second term denies them a proper Lorentz transformation. On the other hand, if one is interested in the three-momentum of the nucleon only, the second term in Eq. (9) drops out and three components of the matrix elements transform just like those of a four-vector. Because Aq,g(μ) are Lorentz scalars, one concludes that the fractions of the nucleon three-momentum carried by quarks and gluons are invariant under Lorentz tranformations! Such a statement, although drawn for non-Lorentz-covariant quantities, does carry important physical significance. Phenomenologically, Aq,g(μ) have been extracted from the parton distributions which have simple interpretations only in the infinite momentum frame. According to the above discussion, the fractions of the nucleon mometum carried by quarks and gluons are also the same in ordinary frames. In particular, if a nucleon has a momentum of 1 GeV/c, then according to the recent analysis [2], roughly 420 MeV/c is carried by gluons in the form of the Poynting vector ∫ d3x~ E × ~ B in the MS scheme and at μ = 1.6 GeV. A more intriguing example concerns the spin structure of the nucleon, which depends on the QCD angular momentum density. The total density M is a mixed Lorentz tensor, expressible in terms of the energy-momentum density T μν in the Belinfante form [3,4], M = T x − T x . (10) The relevant charges J = ∫ dxM are the usual Lorentz generators (including the angular momentum operator ~ J), which tranform as the Lorentz tensor (1, 0) + (0, 1). According to Eq. (10), the angular momentum density has both quark and gluon contributions, M = M q +M μαβ g , where, M q,g = T μβ q,gx α − T μα q,g x . (11) Furthermore, the quark part contains the spin and orbital contributions, M q = M μαβ qL + M qS , where M qS = i 4 ψ̄γ[γ, γ]ψ , M qL = ψ̄γ (xiD − xiD)ψ . (12) Accordingly, the QCD angular momentum operator can be written as a sum of three gaugeinvariant contributions, ~ J = ∫