Analyzed is the effective potential of D-dimensional thermodynamic Gross-Neveu model (defined by large-N leading order, 2 ≤ D < 4) under constant electromagnetic field ( E · B = 0). The potential is derived from a thermal analogy of the worldline formalism. In the magnetic case, the potential is expressed in terms of a discretized momentum sum instead of a continuum momentum integral, and it reproduces known critical values of µ and T in the continuum limit. In the electric case, chiral symmetry is restored at finite T (with arbitrary µ) against instability of fermion vacuum. An electromagnetic duality is pointed out as well. Phase diagrams are obtained in both cases.Our crucial interest is therefore the question how universal the effect of magnetic catalyst on phase structures is in the setting of both finite temperature T and chemical potential µ in various dimensions. We study this question for an electric background field as well. In order to deal with these models all together, it is, of course, useful to analyze a dimensionally regulated effective potential. Such approaches were done in the cases of finite temperature [19], curvature [15] and their mixture [17]. We shall derive a universal expression of the effective potential as a function of D, T , µ and the external field parameter ξ (= e | B| 2 − | E| 2 , where we have assumed B · E = 0) within the leading order of the 1/N-expansion. We shall then present various interesting properties which can be derived from the universal effective potential. Although we have not completed our analyses yet, we confirm that our present results will be useful in order to establish a foundation for further developments in the various branches mentioned above.This paper is organized as follows. In Sect.2, we present a simple and convenient derivation of the effective potential. There is no manifestation of exact fermion propagators of finite temperature field theory in the derivation. Namely, we do not use its Feynman rules to evaluate a fermion loop. Indeed, we apply an analogy of the worldline formalism [22],[23] to the finite temperature cases. Then we write down the proper-time integral representations for the critical equations which determine the phase structures in the ξ-T -µ space.In Sect.3, we discuss the cases of magnetic field dominant, i.e., ξ is a real number.In the former half of this section, we analyze the zero-temperature limit of the effective potentials for finite ξ and µ. The potential is shown to be a discretized version of the one presented in [19]. Also, first order phase transitions are observed in large µ regions on the ξ-µ (T = 0) plane. In the latter half of the section, we discuss the phase structures. We find that a constant magnetic field works as the catalyst in 3 ≤ D < 4 at finite T and µ, and verify that the magnitude of catalyst effect becomes smaller as D or µ increasing.In Sect.4, we present the electric dominant phase diagrams, i.e., the cases of ξ a pure imaginary number. Analyzing the imaginary part of the effective potential, we sh...

The complete set of twist-3 flavor singlet operators which contribute to the structure function g2(X, Q2) is constructed. We develop a covariant approach to derive the anomalous dimension of these operators. We clarify the role played by the operators proportional to the equations of motion and the "alien" operators.The structure function 92(X, Q2) plays a distinguished role in spin physics 1) because it contains information inaccessible by the more familiar spin structure function 91 (x, Q2). 2), 3) In the framework of the operator product expansion (OPE), not only the twist-2 but also the twist-3 operators contribute to 92(X, Q2) to leading order of .)

We numerically analyse the evolution of the flavor non-singlet g 1 structure function taking into account the all-order resummation of α s ln 2 x terms which is expected to have much stronger effects than the DGLAP evolution in the small x region. We include a part of the next-to-leading logarithmic corrections coming from the resummed "coefficient function" which are not considered in the calculation of Blümlein and Vogt to respect the factorization scheme independence. It is pointed out that the resummed coefficient function gives unexpectedly large suppression factor over the experimentally accessible range of x and Q 2 . This fact implies that the next-to-leading logarithmic contributions are very important for the g 1 structure function.

We illustrate the phase structure of a deformed two-dimensional Gross–Neveu model which is defined by undeformed field contents plus deformed Pauli matrices. This deformation is based on two motives to find a more general polymer model and to estimate how q-deformed field theory affects on its effective potential. Some regions where chiral symmetry breaking and restoration take place repeatedly as temperature increasing are found.