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We study the effect of a momentum ( k ) lattice as a regulator of quantum field theory. An an example, we compute the vacuum polarization in noncompact (linearized) QED from k-lattice perturbation theory to one-loop order and study the continuum limit. The amplitude has a finite part plus logarithmically, linearly, and quadratically divergent terms. The amplitude violates gauge invariance (Ward identity) and Lorentz (Euclidean) invariance and is nonlocal. For example, the linear term -~l k is nonlocal. Renormalization requires nonlocal counterterms, which is not inconsistent because the original action on the k lattice already has a nonlocality. We explicitly give the counterterms, which render the amplitude Lorentz and gauge invariant to recover the standard result. PACS number(s): 11.15.Ha, 12.20.D~
We study the effect of a momentum ( k ) lattice as a regulator of quantum field theory. An an example, we compute the vacuum polarization in noncompact (linearized) QED from k-lattice perturbation theory to one-loop order and study the continuum limit. The amplitude has a finite part plus logarithmically, linearly, and quadratically divergent terms. The amplitude violates gauge invariance (Ward identity) and Lorentz (Euclidean) invariance and is nonlocal. For example, the linear term -~l k is nonlocal. Renormalization requires nonlocal counterterms, which is not inconsistent because the original action on the k lattice already has a nonlocality. We explicitly give the counterterms, which render the amplitude Lorentz and gauge invariant to recover the standard result. PACS number(s): 11.15.Ha, 12.20.D~
We suggest a Hamiltonian formulation on a momentum lattice using a physically motivated regularization using the Breit-frame which links the maximal parton number to the lattice size. This scheme restricts parton momenta to positive values in each spatial direction. This leads to a drastic reduction of degrees of freedom compared to a regularization in the rest frame (center at zero momentum). We discuss the computation of physical observables like (i) mass spectrum in the critical region, (ii) structure and distribution functions, (iii) S-matrix, (iv) nite temperature and nite density thermodynamics in the Breit-frame regularization. For the scalar 4 3+1 theory we present numerical results for the mass spectrum in the critical region. We observe scaling behavior for the mass of the ground state and for some higher lying states. We compare our results with renormalization group results by L uscher and Weisz. Using the Breit-frame, we calculate for QCD the relation between the W tensor, structure functions (polarized and unpolarized) and quark distribution functions. We use the improved parton-model with a scale dependence and take into account a non-zero parton mass. In the Bjorken limes we nd the standard relations between F1, F2, g1 and the quark distribution functions. We discuss the rôle of helicity. We present numerical results for parton distribution functions in the scalar model. For the 4 -model we nd no bound state with internal parton structure. For the 3 -model we nd a distribution function with parton structure similar to Altarelli-Parisi behavior of QCD.
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