A factorization algorithm for a patron shower model based on the evolution of momentum distributions proposed in a previous work is studied. The scaling violation of initial state parton distributions is generated using parton showers to an accuracy of the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD) using the information from only splitting functions and initial parton distributions at some fixed low energy. In the algorithm proposed in this paper, the total momentum of the initial state partons is conserved in any factorization scheme for mass singularities. As an example, the scaling violation of the parton distributions and the transverse momentum distributions due to initial state parton radiation are calculated with the jet calculs scheme.

We investigate the Kaluza-Klein (KK) spectrum of N = 1 supersymmetric gauge theory compactified on a circle. We concentrate on a model with gauge group SU (2) and four massless matter fields in the fundamental representation. We derive the exact mass formula of KK modes by using Seiberg-Witten theory. From the mass formula and the D3-brane probe realization, we determine the spectrum of KK modes of matter fields and gauge fields. As a result, we find that the lightest KK state of gauge fields is stable for all the vacuum moduli space, while the lightest KK state of matter fields decays easier than other KK states in a region of the moduli space. The region becomes small as we decrease the five-dimensional gauge coupling constant g 5 , and vanishes as we take the limit g 5 → 0. This result continuously connects the known KK spectrum in the weak coupling limit and that in the strong coupling limit.

We investigate thermal leptogenesis scenarios in the left-right symmetric extension of the standard model. Imposing the realization of D-parity below the GUT scale and grand unification make our model more restrictive and improve its predictive capability. In this case, a D-parity odd singlet plays a critical role. This singlet may cause a very large mass hierarchy of the SU (2) L,R triplet scalars. We test our model by investigating baryogenesis via leptogenesis. Our model has two sources for the lepton number asymmetry in the universe, the heavy right-handed neutrinos N i (i = 1, 2, 3) and the SU (2) L triplet scalar ∆ L . Leptogenesis scenarios can be categorized according to these mass scales. If the light neutrinos are Majorana and have a hierarchical mass spectrum, we can obtain a successful result in leptogenesis through N 1 -decay. However, we find that the normal mass hierarchy of the light neutrinos is inconsistent with leptogenesis through ∆ L -decay in the SM. In order to obtain successful thermal leptogenesis through ∆ Ldecay, we need to introduce additional Higgs doublets. This result suggests the two-Higgs doublet model with an SU (2) L triplet scalar. §1. IntroductionThe SO(10) gauge theory is regarded as a strong possibility for the grand unification theory (GUT). The first important feature of this theory is the unification of three gauge interactions. Because of this feature, the Gell-Mann-Nishijima relation can be rewritten as Q = I 3 L + I 3 R + (B − L)/2; i.e. the electric charge can be quantized and related to classically familiar charges, the baryon number B and the lepton number L. Here I 3 L and I 3 R denote the third components of the left-handed and right-handed isospins, respectively. The second important feature of this theory is matter unification. In contrast to SU (5) GUT, in the SO(10) gauge theory, each quark and lepton corresponds to a 5-bit eigenstate of the Cartan subalgebra: 1)where ↑ and ↓ in the kets denote the eigenvalues of the five generators of the Cartan subalgebra: The first three arrows are for SO (6) SU (4) c , and last two are for SO(4) SU (2) L × SU (2) R . 2) Consequently, by adding the three right-handed neutrinos, we can obtain an economic picture of matter in the universe. * )